# elliott

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```					     Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
Rings of Integer-Valued Polynomials
Integer-valued
polynomial rings

Krull domains
and PVMDs                                   Jesse Elliott
Universal                 California State University, Channel Islands
properties of IVP
rings

Summmary of
open problems

Number Theory Seminar
UCLA
February 25, 2008

UCLA NTS 02/25/08                   [1]                            jesse.elliott@csuci.edu
Outline

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction          1   Introduction
Prüfer domains

Integer-valued
polynomial rings
2   Prüfer domains
Krull domains
and PVMDs             3   Integer-valued polynomial rings
Universal
properties of IVP
rings                 4   Krull domains and PVMD’s
Summmary of
open problems         5   Universal properties of IVP rings
6   Summary of open problems

UCLA NTS 02/25/08                           [2]             jesse.elliott@csuci.edu
Integer-valued polynomials (IVP’s)

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [3]                        jesse.elliott@csuci.edu
Integer-valued polynomials (IVP’s)

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings      Example: The binomial coefﬁcient polynomial
Krull domains
and PVMDs
X        X(X − 1)(X − 2) · · · (X − n + 1)
Universal                                =
properties of IVP
rings
n                     n!
Summmary of
open problems         is integer-valued for every positive integer n.

UCLA NTS 02/25/08                             [3]                         jesse.elliott@csuci.edu
Integer-valued polynomials (IVP’s)

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A polynomial f (X) ∈ Q[X] is said to be integer-valued if
Introduction
f (Z) ⊆ Z.
Prüfer domains

Integer-valued
polynomial rings      Example: The binomial coefﬁcient polynomial
Krull domains
and PVMDs
X        X(X − 1)(X − 2) · · · (X − n + 1)
Universal                                =
properties of IVP
rings
n                     n!
Summmary of
open problems         is integer-valued for every positive integer n.
p
−X
Example: The Fermat binomial X p is integer-valued for any
prime number p (⇐⇒ Fermat’s little theorem)

UCLA NTS 02/25/08                             [3]                         jesse.elliott@csuci.edu
Describing all IVP’s

Rings of
Integer-Valued
Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
Jesse Elliott
Q[X].
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
Describing all IVP’s

Rings of
Integer-Valued
Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
Jesse Elliott
Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
composition.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
Describing all IVP’s

Rings of
Integer-Valued
Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
Jesse Elliott
Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
composition.
Krull domains
and PVMDs
Int(Z) is generated freely as a Z-module by the binomial
Universal
properties of IVP     coefﬁcient polynomials:
rings

Summmary of                                             ∞
open problems                                                 X
Int(Z) =           Z.
n=0
n

UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
Describing all IVP’s

Rings of
Integer-Valued
Polynomials        Let Int(Z) denote the set of all integer-valued polynomials in
Jesse Elliott
Q[X].
Introduction

Prüfer domains        Int(Z) is a subring of Q[X] containing Z[X] that is closed under
Integer-valued
polynomial rings
composition.
Krull domains
and PVMDs
Int(Z) is generated freely as a Z-module by the binomial
Universal
properties of IVP     coefﬁcient polynomials:
rings

Summmary of                                             ∞
open problems                                                 X
Int(Z) =           Z.
n=0
n

Int(Z) is generated, as a subring of Q[X] closed under
p
−X
composition, by X and the Fermat binomials X p for p prime.

UCLA NTS 02/25/08                            [4]                         jesse.elliott@csuci.edu
IVP’s and prime numbers

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                 [5]       jesse.elliott@csuci.edu
IVP’s and prime numbers

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem (Tao and Ziegler, 2006)
Prüfer domains

Integer-valued
Given any integer-valued polynomials P1 , P2 , . . . , Pk with vanish-
polynomial rings
ing constant terms, there are inﬁnitely many integers x and m
Krull domains
and PVMDs             such that the integers x+P1 (m), . . . , x+Pk (m) are simultaneously
Universal             prime.
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [5]                          jesse.elliott@csuci.edu
IVP’s and prime numbers

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem (Tao and Ziegler, 2006)
Prüfer domains

Integer-valued
Given any integer-valued polynomials P1 , P2 , . . . , Pk with vanish-
polynomial rings
ing constant terms, there are inﬁnitely many integers x and m
Krull domains
and PVMDs             such that the integers x+P1 (m), . . . , x+Pk (m) are simultaneously
Universal             prime.
properties of IVP
rings

Summmary of
open problems         The special case when the polynomials are m, 2m, . . . , km implies
that there are arithmetic progressions of primes of length k for
any k.

UCLA NTS 02/25/08                            [5]                          jesse.elliott@csuci.edu
Int(Z)

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
Int(Z)

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings
However, Int(Z) is not Noetherian. The ideal XQ[X] ∩ Int(Z) of
Krull domains
and PVMDs             polynomials with vanishing constant term is not ﬁnitely generated.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
Int(Z)

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
Like Z[X], the ring Int(Z) is integrally closed of Krull dimension 2.
Integer-valued
polynomial rings
However, Int(Z) is not Noetherian. The ideal XQ[X] ∩ Int(Z) of
Krull domains
and PVMDs             polynomials with vanishing constant term is not ﬁnitely generated.
Universal
properties of IVP
rings                 Int(Z) is like a “non-Noetherian Dedekind domain” in that the
Summmary of
open problems
nonzero ﬁnitely generated ideals of Int(Z) are invertible and
2-generated.

UCLA NTS 02/25/08                            [6]                          jesse.elliott@csuci.edu
Spec(Int(Z))

Rings of
Integer-Valued       One has
Polynomials

Jesse Elliott
Spec(Q[X]) ←→   {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
q(X)Q[X] −→    pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                     [7]                        jesse.elliott@csuci.edu
Spec(Int(Z))

Rings of
Integer-Valued       One has
Polynomials

Jesse Elliott
Spec(Q[X]) ←→     {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
q(X)Q[X] −→      pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
Zp   ←→   {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
α   −→   mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems

UCLA NTS 02/25/08                        [7]                       jesse.elliott@csuci.edu
Spec(Int(Z))

Rings of
Integer-Valued       One has
Polynomials

Jesse Elliott
Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .

UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
Spec(Int(Z))

Rings of
Integer-Valued       One has
Polynomials

Jesse Elliott
Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .

max-Spec(Int(Z)) ←→        Zp
p

UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
Spec(Int(Z))

Rings of
Integer-Valued       One has
Polynomials

Jesse Elliott
Spec(Q[X]) ←→         {prime ideals of Int(Z) above (0) ⊂ Z}
Introduction
q(X)Q[X] −→          pq(X) = q(X)Q[X] ∩ Int(Z).
Prüfer domains

Integer-valued
polynomial rings      One has
Krull domains
and PVMDs
Zp   ←→     {prime ideals of Int(Z) above (p) ⊂ Z}
Universal
properties of IVP
rings
α   −→     mp,α = {f (X) ∈ Int(Z) : f (α) ∈ pZp }
Summmary of
open problems         mp,α ⊇ pq(X) iff q(α) = 0 in Qp .

max-Spec(Int(Z)) ←→        Zp
p

mp,α has height 2 iff α ∈ Zp is algebraic over Q.

mp,α has height 1 iff α ∈ Zp is transcendental over Q.
UCLA NTS 02/25/08                             [7]                       jesse.elliott@csuci.edu
Stone-Weierstrass

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains        Any continuous function f : Zp −→ Zp can be expanded uniquely
Integer-valued
polynomial rings
as
∞
Krull domains
X
and PVMDs                                     f=       an
n=0
n
Universal
properties of IVP
rings                 with an ∈ Zp for all n and limn an = 0.
Summmary of
open problems

UCLA NTS 02/25/08                            [8]                    jesse.elliott@csuci.edu
Stone-Weierstrass

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains        Any continuous function f : Zp −→ Zp can be expanded uniquely
Integer-valued
polynomial rings
as
∞
Krull domains
X
and PVMDs                                     f=       an
n=0
n
Universal
properties of IVP
rings                 with an ∈ Zp for all n and limn an = 0.
Summmary of
open problems
Int(Z) is dense in C(Zp , Zp ).

UCLA NTS 02/25/08                             [8]                   jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [9]                       jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains
1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [9]                          jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains
1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [9]                        jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains
1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
prime ideal p of D.
Summmary of
open problems

UCLA NTS 02/25/08                           [9]                       jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains
1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
prime ideal p of D.
Summmary of
open problems
4   D is Noetherian integrally closed of Krull dimension at most
1.

UCLA NTS 02/25/08                           [9]                        jesse.elliott@csuci.edu
Dedekind domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Recall that a domain D is a Dedekind domain if any of the
following equivalent conditions holds.
Introduction

Prüfer domains
1   Every nonzero ideal of D is invertible (or projective).
Integer-valued
polynomial rings
2   The set of nonzero fractional ideals of D is a group under
Krull domains              multiplication.
and PVMDs

Universal
3   D is Noetherian, and Dp is a DVR (or PID) for every nonzero
properties of IVP
rings
prime ideal p of D.
Summmary of
open problems
4   D is Noetherian integrally closed of Krull dimension at most
1.

The ideal class group of D Dedekind is the group of nonzero
fractional ideals of D modulo the subgroup of nonzero principal
fractional ideals of D.

UCLA NTS 02/25/08                           [9]                        jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [10]                      jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains
I(D) is the group of nonzero invertible (or projective)
Integer-valued
fractional ideals of D
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains
I(D) is the group of nonzero invertible (or projective)
Integer-valued
fractional ideals of D
polynomial rings

Krull domains
P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains
I(D) is the group of nonzero invertible (or projective)
Integer-valued
fractional ideals of D
polynomial rings

Krull domains
P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems

UCLA NTS 02/25/08                           [10]                        jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains
I(D) is the group of nonzero invertible (or projective)
Integer-valued
fractional ideals of D
polynomial rings

Krull domains
P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems

Invertible ideals are necessarily ﬁnitely generated. If the converse
is true in a domain D, we say that D is a Prüfer domain (Prüfer
1932, Krull 1936).

UCLA NTS 02/25/08                           [10]                         jesse.elliott@csuci.edu
Picard group

Rings of
Integer-Valued
Polynomials
The Picard group Pic(D) of a domain D is the group
Jesse Elliott
I(D)/P(D), where
Introduction

Prüfer domains
I(D) is the group of nonzero invertible (or projective)
Integer-valued
fractional ideals of D
polynomial rings

Krull domains
P(D) is the subgroup of nonzero principal (or free) fractional
and PVMDs                 ideals of D.
Universal
properties of IVP
rings

Summmary of
D Dedekind domain =⇒ Pic(D) = the ideal class group of D.
open problems

Invertible ideals are necessarily ﬁnitely generated. If the converse
is true in a domain D, we say that D is a Prüfer domain (Prüfer
1932, Krull 1936).

D Dedekind ⇐⇒ D Noetherian and Prüfer.

UCLA NTS 02/25/08                           [10]                         jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [11]                       jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero ﬁnitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero ﬁnitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [11]                        jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero ﬁnitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefﬁcients is a unit
of D.

UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero ﬁnitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefﬁcients is a unit
of D.
5   Every submodule of a projective D-module is projective.

UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A domain D with quotient ﬁeld K is a Prüfer domain iff any of the
Introduction          following equivalent conditions holds.
Prüfer domains
1   Every nonzero ﬁnitely generated ideal of D is invertible.
Integer-valued
polynomial rings       2   The set of nonzero ﬁnitely generated fractional ideals of D is
Krull domains
and PVMDs                  a group under multiplication.
Universal
properties of IVP
3   Dp is a valuation domain for every prime ideal p of D.
rings

Summmary of
4   D is integrally closed, and every element of K is a root of a
open problems              polynomial in D[X] at least one of whose coefﬁcients is a unit
of D.
5   Every submodule of a projective D-module is projective.
6   Every D-torsionfree D-module is D-ﬂat.

UCLA NTS 02/25/08                           [11]                        jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]    jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers.
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [12]                     jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
algebraic extension of its quotient ﬁeld.

UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
algebraic extension of its quotient ﬁeld.
D[X] Prüfer ⇐⇒ D ﬁeld.

UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
Prüfer domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Prüfer domains:
Introduction
Any Dedekind domain.
Prüfer domains

Integer-valued
Int(Z)
polynomial rings

Krull domains
Any Bézout domain, i.e. a domain every ﬁnitely generated
and PVMDs                ideal of which is principal.
Universal
properties of IVP        The ring A of algebraic integers. A is a Bézout domain.
rings

Summmary of              The integral closure of any Dedekind or Prüfer domain in any
open problems
algebraic extension of its quotient ﬁeld.
D[X] Prüfer ⇐⇒ D ﬁeld.

D is a Bézout domain ⇐⇒ D is a Prüfer domain and Pic(D) = 0.

UCLA NTS 02/25/08                          [12]                      jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
A non-Noetherian Prüfer of Krull dimension 1, every ﬁnitely
Integer-valued
generated ideal principal.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
A non-Noetherian Prüfer of Krull dimension 1, every ﬁnitely
Integer-valued
generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every ﬁnitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
A non-Noetherian Prüfer of Krull dimension 1, every ﬁnitely
Integer-valued
generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every ﬁnitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems
D Prüfer of Krull dimension n =⇒ every ﬁnitely generated ideal of
D can be generated by n + 1 elements.

UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
Prüfer domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
A non-Noetherian Prüfer of Krull dimension 1, every ﬁnitely
Integer-valued
generated ideal principal.
polynomial rings

Krull domains
and PVMDs             Int(Z) non-Noetherian Prüfer of Krull dimension 2, every ﬁnitely
Universal             generated ideal 2-generated.
properties of IVP
rings

Summmary of
open problems
D Prüfer of Krull dimension n =⇒ every ﬁnitely generated ideal of
D can be generated by n + 1 elements. This is the best result
possible for arbitrary Prüfer domains.

UCLA NTS 02/25/08                           [13]                       jesse.elliott@csuci.edu
Generalized IVP’s

Rings of
Integer-Valued
Polynomials        Let D an integral domain with quotient ﬁeld K. The ring of
Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
of K[X].
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [14]                       jesse.elliott@csuci.edu
Generalized IVP’s

Rings of
Integer-Valued
Polynomials        Let D an integral domain with quotient ﬁeld K. The ring of
Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
of K[X].
Krull domains
and PVMDs

Universal             Int(D) was ﬁrst studied, for number ﬁelds D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems

UCLA NTS 02/25/08                          [14]                      jesse.elliott@csuci.edu
Generalized IVP’s

Rings of
Integer-Valued
Polynomials        Let D an integral domain with quotient ﬁeld K. The ring of
Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
of K[X].
Krull domains
and PVMDs

Universal             Int(D) was ﬁrst studied, for number ﬁelds D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems
D Dedekind =⇒ Int(D) is free as a D-module . . . but a basis may
be hard to compute.

UCLA NTS 02/25/08                          [14]                       jesse.elliott@csuci.edu
Generalized IVP’s

Rings of
Integer-Valued
Polynomials        Let D an integral domain with quotient ﬁeld K. The ring of
Jesse Elliott       (generalized) integer-valued polynomials on D is the subring
Introduction

Prüfer domains                       Int(D) = {f ∈ K[X] | f (D) ⊆ D}
Integer-valued
polynomial rings
of K[X].
Krull domains
and PVMDs

Universal             Int(D) was ﬁrst studied, for number ﬁelds D, by Pólya and
properties of IVP
rings                 Ostrowski circa 1917. They sought a D-module basis for Int(D).
Summmary of
open problems
D Dedekind =⇒ Int(D) is free as a D-module . . . but a basis may
be hard to compute.

D Dedekind domain with ﬁnite residue ﬁelds =⇒ Int(D) is a
non-Noetherian Prüfer domain of Krull dimension 2 with every
ﬁnitely generated ideal 2-generated.

UCLA NTS 02/25/08                          [14]                       jesse.elliott@csuci.edu
IVP’s over DVR’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
D DVR with ﬁnite residue ﬁeld D/m =⇒ Int(D) is dense in
Integer-valued        C(Dm , Dm ) for the uniform convergence topology.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [15]                      jesse.elliott@csuci.edu
IVP’s over DVR’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
D DVR with ﬁnite residue ﬁeld D/m =⇒ Int(D) is dense in
Integer-valued        C(Dm , Dm ) for the uniform convergence topology.
polynomial rings

Krull domains
and PVMDs             In fact, Int(D) (= D[X]) has a D-basis 1, X, f2 (X), f3 (X), . . .
Universal
properties of IVP     such that every continuous function ϕ : Dm −→ Dm can be written
rings
uniquely in the form
Summmary of                                               ∞
open problems
ϕ=         ai fi ,
i=0

where ai ∈ Dm and limi ai = 0.

UCLA NTS 02/25/08                           [15]                        jesse.elliott@csuci.edu
An example: Int(Z[i])

Rings of
Integer-Valued
Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
2
Jesse Elliott       Int(Z) is not a subset of Int(Z[i]).
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [16]                        jesse.elliott@csuci.edu
An example: Int(Z[i])

Rings of
Integer-Valued
Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
2
Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [16]                          jesse.elliott@csuci.edu
An example: Int(Z[i])

Rings of
Integer-Valued
Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
2
Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains
A result of Pólya and Ostrowski yields a Z[i]-basis
Integer-valued
4
2        3   2       −2X 3 −iX 2 +(1+i)X (5+(1+i)4 )X 5 +···
1, X, X1+i , X 1+i , X
−X      −X
polynomial rings
(1+i)3        ,    5(1+i)3         ,...
Krull domains
and PVMDs

Universal
properties of IVP     for Int(Z[i]).
rings

Summmary of
open problems

UCLA NTS 02/25/08                                 [16]                             jesse.elliott@csuci.edu
An example: Int(Z[i])

Rings of
Integer-Valued
Polynomials        Note that X lies in Int(Z) but not in Int(Z[i]) and therefore
2
Jesse Elliott       Int(Z) is not a subset of Int(Z[i]). Int(D) is not functorial in D.
Introduction

Prüfer domains
A result of Pólya and Ostrowski yields a Z[i]-basis
Integer-valued
4
2         3     2       −2X 3 −iX 2 +(1+i)X (5+(1+i)4 )X 5 +···
1, X, X1+i , X 1+i , X
−X      −X
polynomial rings
(1+i)3        ,    5(1+i)3         ,...
Krull domains
and PVMDs

Universal
properties of IVP     for Int(Z[i]).
rings

Summmary of
open problems         Int(Z[i]) is generated under the D-algebra operations and
composition by the so-called “Fermat binomials”
X 2 −X
F2 =    1+i
X p −X
Fp =      p      for p ≡ 1 (mod 4)
p2
X     −X
Fp2 =        p     for p ≡ 3 (mod 4).

UCLA NTS 02/25/08                                    [16]                             jesse.elliott@csuci.edu
Bases

Rings of
Integer-Valued
Polynomials

Jesse Elliott
(Pólya and Ostrowski) If D is a number ring (or any Dedekind
Introduction          domain) such that
Prüfer domains
Πq =       m
Integer-valued
polynomial rings                                      N (m)=q
Krull domains
and PVMDs             is principal for all prime powers q, then Int(D) is generated under
Universal             the D-algebra operations and composition by X and the
properties of IVP
rings                 binomials
Summmary of                                         Xq − X
open problems                                  Fq =          for all q,
πq
where Πq = πq D for all q.

UCLA NTS 02/25/08                           [17]                        jesse.elliott@csuci.edu
Bases

Rings of
Integer-Valued
Polynomials

Jesse Elliott
(Pólya and Ostrowski) If D is a number ring (or any Dedekind
Introduction          domain) such that
Prüfer domains
Πq =       m
Integer-valued
polynomial rings                                      N (m)=q
Krull domains
and PVMDs             is principal for all prime powers q, then Int(D) is generated under
Universal             the D-algebra operations and composition by X and the
properties of IVP
rings                 binomials
Summmary of                                         Xq − X
open problems                                  Fq =          for all q,
πq
where Πq = πq D for all q.

In that case they also constructed a D-basis for Int(D).

UCLA NTS 02/25/08                           [17]                        jesse.elliott@csuci.edu
Int(D) as a D-module

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Problem
Introduction

Prüfer domains
Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [18]                       jesse.elliott@csuci.edu
Int(D) as a D-module

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Problem
Introduction

Prüfer domains
Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs
Problem
Universal
properties of IVP
rings                 For which domains D is Int(D) ﬂat as a D-module?
Summmary of
open problems

UCLA NTS 02/25/08                          [18]                       jesse.elliott@csuci.edu
Int(D) as a D-module

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Problem
Introduction

Prüfer domains
Find an example, if one exists, of a domain D such that Int(D) is
Integer-valued
not free. For which domains D is Int(D) free as a D-module?
polynomial rings

Krull domains
and PVMDs
Problem
Universal
properties of IVP
rings                 For which domains D is Int(D) ﬂat as a D-module?
Summmary of
open problems

Proposition
Int(D) is locally free (hence faithfully ﬂat) as a D-module for any
“Krull domain” D.

UCLA NTS 02/25/08                           [18]                        jesse.elliott@csuci.edu
More examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
If D is a ﬁeld, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [19]          jesse.elliott@csuci.edu
More examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
If D is a ﬁeld, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs
Int(D) might equal D[X] even if D is not a ﬁeld. This happens, for
Universal
example, if D = D [T ] and D is not a ﬁnite ﬁeld, or if D/m is
properties of IVP
rings
inﬁnite for every maximal ideal m of D.
Summmary of
open problems

UCLA NTS 02/25/08                           [19]                       jesse.elliott@csuci.edu
More examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
If D is a ﬁeld, then Int(D) = D[X].
Integer-valued
polynomial rings

Krull domains
and PVMDs
Int(D) might equal D[X] even if D is not a ﬁeld. This happens, for
Universal
example, if D = D [T ] and D is not a ﬁnite ﬁeld, or if D/m is
properties of IVP
rings
inﬁnite for every maximal ideal m of D.
Summmary of
open problems

UCLA NTS 02/25/08                           [19]                       jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Generally, for any set X, we let
Introduction

Prüfer domains
Int(DX ) = {f (X) ∈ F [X] : f (DX ) ⊆ D}.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [20]                       jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Generally, for any set X, we let
Introduction

Prüfer domains
Int(DX ) = {f (X) ∈ F [X] : f (DX ) ⊆ D}.
Integer-valued
polynomial rings

Krull domains
and PVMDs
Just as
Universal
properties of IVP                            (R[X])[Y ] = R[X       Y]
rings

Summmary of
open problems
for any commutative ring R, one has

Int(Int(DX )Y ) = Int(DX        Y
),

as long as D is inﬁnite (not a ﬁnite ﬁeld).

UCLA NTS 02/25/08                            [20]                               jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
θX :           Int(D) −→ Int(DX ),
polynomial rings                              T ∈X
Krull domains
and PVMDs
where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [21]                         jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
θX :           Int(D) −→ Int(DX ),
polynomial rings                               T ∈X
Krull domains
and PVMDs
where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
If Int(D) = D[X] (e.g. if D is a ﬁeld), then θX is an isomorphism
open problems         for every set X.

UCLA NTS 02/25/08                             [21]                         jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
For any set X, one has a canonical D-algebra homomorphism
Introduction

Prüfer domains

Integer-valued
θX :           Int(D) −→ Int(DX ),
polynomial rings                               T ∈X
Krull domains
and PVMDs
where the tensor product is over D.
Universal
properties of IVP
rings

Summmary of
If Int(D) = D[X] (e.g. if D is a ﬁeld), then θX is an isomorphism
open problems         for every set X.

If Int(D) is ﬂat as a D-module (e.g if D is a Prüfer domain), then
θX is injective for every set X.

UCLA NTS 02/25/08                             [21]                         jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction

Prüfer domains        If D is a “Krull domain,” or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs
for every set X.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [22]                      jesse.elliott@csuci.edu
Multivariable integer-valued polynomials

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction

Prüfer domains        If D is a “Krull domain,” or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs
for every set X.
Universal
properties of IVP
rings

Summmary of           Problem
open problems
Find an example, if one exists, of a domain D and set X for which
θX is not surjective/injective/an isomorphism. For which domains
D is θX an isomorphism for all sets X?

UCLA NTS 02/25/08                          [22]                       jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
“Krull.”
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [23]                       jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
“Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
discrete valuations on the fraction ﬁeld K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.

UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
“Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
discrete valuations on the fraction ﬁeld K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.

Every Krull domain is integrally closed.

UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
The integral closure of a Noetherian domain of Krull dimension
Introduction          greater than 2 need not be Noetherian. However, it must be
Prüfer domains
“Krull.”
Integer-valued
polynomial rings

Krull domains         A domain D is a Krull domain if there exists a family {vi } of
and PVMDs
discrete valuations on the fraction ﬁeld K of D such that
Universal
properties of IVP
rings                      For all x ∈ K ∗ , vi (x) = 0 for almost all i.
Summmary of
open problems
For all x ∈ K ∗ , x ∈ D iff vi (x) ≥ 0 for all i.

Every Krull domain is integrally closed.

D Dedekind ⇐⇒ D Krull of Krull dimension 1.

UCLA NTS 02/25/08                           [23]                        jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [24]   jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains
Any UFD or integrally closed Noetherian domain.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [24]                      jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains
Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [24]                      jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains
Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
ﬁnite extension of its quotient ﬁeld.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains
Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
ﬁnite extension of its quotient ﬁeld.
Universal
properties of IVP
rings
D[X] Krull ⇐⇒ D Krull.
Summmary of
open problems

UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
Krull domains: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of Krull domains:
Introduction

Prüfer domains
Any UFD or integrally closed Noetherian domain.
Integer-valued           The integral closure of any Noetherian domain.
polynomial rings

Krull domains            The integral closure of any Noetherian or Krull domain in any
and PVMDs
ﬁnite extension of its quotient ﬁeld.
Universal
properties of IVP
rings
D[X] Krull ⇐⇒ D Krull.
Summmary of
open problems

Bouvier’s Conjecture (1985)
There exists a UFD or Krull domain D such that dim D[X] > 1 +
dim D.

UCLA NTS 02/25/08                          [24]                       jesse.elliott@csuci.edu
v-class group

Rings of
Integer-Valued
Polynomials
A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
Jesse Elliott
be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [25]                         jesse.elliott@csuci.edu
v-class group

Rings of
Integer-Valued
Polynomials
A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
Jesse Elliott
be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [25]                         jesse.elliott@csuci.edu
v-class group

Rings of
Integer-Valued
Polynomials
A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
Jesse Elliott
be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
or, equivalently, if Iv is a unit in Iv (D).
open problems

UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
v-class group

Rings of
Integer-Valued
Polynomials
A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
Jesse Elliott
be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
or, equivalently, if Iv is a unit in Iv (D).
open problems

The v-class group Clv (D) = the group of v-invertible divisorial
fractional ideals under v-multiplication modulo the subgroup of
nonzero principal fractional ideals.

UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
v-class group

Rings of
Integer-Valued
Polynomials
A fractional ideal I of a domain D such that I = (I −1 )−1 is said to
Jesse Elliott
be divisorial. The divisorial closure Iv = (I −1 )−1 of a fractional
Introduction          ideal I is necessarily divisorial.
Prüfer domains

Integer-valued
polynomial rings      The divisorial fractional ideals of D form a monoid Iv (D) under
Krull domains         the operation I ·v J = (IJ)v , called v-multiplication.
and PVMDs

Universal
properties of IVP     A fractional ideal I of D is said to be v-invertible if (II −1 )v = D,
rings

Summmary of
or, equivalently, if Iv is a unit in Iv (D).
open problems

The v-class group Clv (D) = the group of v-invertible divisorial
fractional ideals under v-multiplication modulo the subgroup of
nonzero principal fractional ideals.

D is a UFD ⇐⇒ D is a Krull domain such that Clv (D) = 0.

UCLA NTS 02/25/08                             [25]                         jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [26]                         jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [26]                          jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
3   D = p∈X 1 (D) Dp , where X 1 (D) is the set of height one
rings

Summmary of
prime ideals of D, where Dp is a DVR for all p ∈ X 1 (D), and
open problems              where every element of D lies in only ﬁnitely many
p ∈ X 1 (D).

UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott
A domain D is a Krull domain if and only if any of the following
Introduction          equivalent conditions holds.
Prüfer domains

Integer-valued
1   Every nonzero divisorial ideal of D is v-invertible.
polynomial rings
2   The set of all nonzero divisorial fractional ideals of D is a
Krull domains
and PVMDs                  group under v-multiplication.
Universal
properties of IVP
3   D = p∈X 1 (D) Dp , where X 1 (D) is the set of height one
rings

Summmary of
prime ideals of D, where Dp is a DVR for all p ∈ X 1 (D), and
open problems              where every element of D lies in only ﬁnitely many
p ∈ X 1 (D).
4   D is a Mori domain, that is, D satisﬁes the ACC on divisorial
ideals, and D is “completely integrally closed.”

UCLA NTS 02/25/08                           [26]                         jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction          D Krull domain ⇐⇒ {nonzero divisorial fractional ideals of D} is
Prüfer domains
an abelian group under v-multiplication, in which case it is free on
Integer-valued
polynomial rings      the set X 1 (D) of prime ideals of height one.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [27]                        jesse.elliott@csuci.edu
Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction          D Krull domain ⇐⇒ {nonzero divisorial fractional ideals of D} is
Prüfer domains
an abelian group under v-multiplication, in which case it is free on
Integer-valued
polynomial rings      the set X 1 (D) of prime ideals of height one.
Krull domains
and PVMDs

Universal
Every nonzero divisorial fractional ideal I in a Krull domain D has
properties of IVP
rings
a unique primary decomposition
Summmary of
open problems
I=                 p(np ) .
p∈X 1 (D)

UCLA NTS 02/25/08                            [27]                          jesse.elliott@csuci.edu
t-class group

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A fractional ideal I of a domain D is said to be v-ﬁnite if Iv = Jv
Introduction          for some ﬁnitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-ﬁnite.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [28]                         jesse.elliott@csuci.edu
t-class group

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A fractional ideal I of a domain D is said to be v-ﬁnite if Iv = Jv
Introduction          for some ﬁnitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-ﬁnite.
Integer-valued
polynomial rings

Krull domains
The v-ﬁnite divisorial fractional ideals of D form a submonoid
and PVMDs
Ivf (D) of Iv (D).
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [28]                         jesse.elliott@csuci.edu
t-class group

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A fractional ideal I of a domain D is said to be v-ﬁnite if Iv = Jv
Introduction          for some ﬁnitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-ﬁnite.
Integer-valued
polynomial rings

Krull domains
The v-ﬁnite divisorial fractional ideals of D form a submonoid
and PVMDs
Ivf (D) of Iv (D).
Universal
properties of IVP
rings
I is said to be t-invertible t-ideal if I is a unit in the monoid
Summmary of
open problems         Ivf (D) (that is, I is a v-invertible divisorial fractional ideal and
both I and I −1 are v-ﬁnite).

UCLA NTS 02/25/08                             [28]                           jesse.elliott@csuci.edu
t-class group

Rings of
Integer-Valued
Polynomials

Jesse Elliott       A fractional ideal I of a domain D is said to be v-ﬁnite if Iv = Jv
Introduction          for some ﬁnitely generated ideal J. A domain D is Mori iff every
Prüfer domains        (divisorial) ideal of D is v-ﬁnite.
Integer-valued
polynomial rings

Krull domains
The v-ﬁnite divisorial fractional ideals of D form a submonoid
and PVMDs
Ivf (D) of Iv (D).
Universal
properties of IVP
rings
I is said to be t-invertible t-ideal if I is a unit in the monoid
Summmary of
open problems         Ivf (D) (that is, I is a v-invertible divisorial fractional ideal and
both I and I −1 are v-ﬁnite).

Clt (D) = group of t-invertible t-ideals mod principal fractional
ideals (= Clv (D) if D is Mori).

UCLA NTS 02/25/08                             [28]                           jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [29]                      jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
1   Every nonzero v-ﬁnite divisorial ideal of D is v-invertible.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
1   Every nonzero v-ﬁnite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-ﬁnite divisorial fractional ideals of D
and PVMDs

Universal
is a group under v-multiplication.
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
1   Every nonzero v-ﬁnite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-ﬁnite divisorial fractional ideals of D
and PVMDs

Universal
is a group under v-multiplication.
properties of IVP
rings
3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems

UCLA NTS 02/25/08                            [29]                          jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
1   Every nonzero v-ﬁnite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-ﬁnite divisorial fractional ideals of D
and PVMDs

Universal
is a group under v-multiplication.
properties of IVP
rings
3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems
4   D is integrally closed and “(I ∩ J)t = It ∩ Jt ” for all ideals I
and J of D.

UCLA NTS 02/25/08                             [29]                         jesse.elliott@csuci.edu
PVMD’s

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
A domain D is said to be a Prüfer v-multiplication domain, or
Prüfer domains
PVMD, if any of the following equivalent conditions holds.
Integer-valued
polynomial rings
1   Every nonzero v-ﬁnite divisorial ideal of D is v-invertible.
Krull domains          2   The set of all nonzero v-ﬁnite divisorial fractional ideals of D
and PVMDs

Universal
is a group under v-multiplication.
properties of IVP
rings
3   Dp is a valuation domain for every “t-maximal ideal” p of D.
Summmary of
open problems
4   D is integrally closed and “(I ∩ J)t = It ∩ Jt ” for all ideals I
and J of D.

D Krull ⇐⇒ D Mori PVMD.

UCLA NTS 02/25/08                             [29]                         jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [30]   jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [30]              jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every ﬁnite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [30]                     jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every ﬁnite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
its quotient ﬁeld.
Summmary of
open problems

UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every ﬁnite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
its quotient ﬁeld.
Summmary of
open problems
D[X] PVMD ⇐⇒ D PVMD.

UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains           Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings         Any GCD domain, a domain in which every ﬁnite set of
Krull domains            elements has a GCD.
and PVMDs

Universal
The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
its quotient ﬁeld.
Summmary of
open problems
D[X] PVMD ⇐⇒ D PVMD.
D Krull =⇒ Int(D) PVMD!

UCLA NTS 02/25/08                         [30]                       jesse.elliott@csuci.edu
PVMD’s: examples

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Examples of PVMD’s:
Introduction

Prüfer domains            Any Krull domain or Prüfer domain.
Integer-valued
polynomial rings          Any GCD domain, a domain in which every ﬁnite set of
Krull domains             elements has a GCD.
and PVMDs

Universal
The integral closure of a PVMD in any algebraic extension of
properties of IVP
rings
its quotient ﬁeld.
Summmary of
open problems
D[X] PVMD ⇐⇒ D PVMD.
D Krull =⇒ Int(D) PVMD!

D is a GCD domain ⇐⇒ D is a PVMD such that Clt (D) = 0.

UCLA NTS 02/25/08                          [30]                       jesse.elliott@csuci.edu
Some classes of integrally closed domains

Rings of
Integer-Valued
Polynomials
Z NNN           DVR            k[X]
Jesse Elliott
NNN                     q
NNN               qqq
Introduction
NNN         qqq
Prüfer domains                               N'  xqqq
Integer-valued                              Euclidean
polynomial rings

Krull domains
and PVMDs

Universal
PID MM
oo         MMM
properties of IVP

ooo
rings

oo                   MMM
o                        MMM
wooo
Summmary of
open problems
             &
DedekindOOO             UFD M          Bézout
OOOooooo          MMM qqq
MMM
ooOOO               qqq
 wo  ooo       OO'     xqqqq MMM& 
Krull O             Prüfer              GCD
OOO
OOO                     qqq
OOO             qqq
O'  xqqqq
UCLA NTS 02/25/08                       [31]
PVMD                      jesse.elliott@csuci.edu
Some classes of integrally closed domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
PID K
Prüfer domains
q        KKK
qqq            KKK
qqq
Integer-valued
KKK
qq
polynomial rings

Krull domains                        xq                         %
and PVMDs
DedekindMMM          UFD K         Bézout
MMMqqqq        KKK ss
Universal
KKKs
ss
properties of IVP
qqqqMMM          sss KKK% 
rings
 xqq         M&     yss
Summmary of
open problems                  Krull M          Prüfer           GCD
MMM                      s
MMM                sss
MMM          sss
&  ysss
PVMD

UCLA NTS 02/25/08                       [32]                           jesse.elliott@csuci.edu
In a larger context

Rings of
Integer-Valued
Polynomials

Jesse Elliott
PID K
qq             KKK
qqq
Introduction
KKK
Prüfer domains
qqq                        KKK
Integer-valued                                                xqqq                              %
polynomial rings
Dedekind                        UFD K             Bézout
n  MMMM qqq                              KKK ss
nnn                      MMMq                   KKKs
Krull domains

nnn                         qqMM                   ss
sss KKK% 
and PVMDs

nnnn                        qqqq MM                s
Universal
properties of IVP                 wn                        xq            &          ys
rings
Noetherian
PPP               n Krull MMMM                Prüfer               GCD
PPP nnnn                            q                     s
Summmary of
P
n                          Mqqqqq
M                     ssss
nnn PPPPP                    qq MMMMM  ssss
open problems
                               xqqq
wnnn           '                            &  ys
Mori PP               Coherent
PVMD
PPP                                 qq
PPP
PPP

qqqqq
P'  xqqqq
v-coherent

UCLA NTS 02/25/08                                 [33]                                   jesse.elliott@csuci.edu
Krull dimension

Rings of
Integer-Valued
Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
Jesse Elliott       hard to compute for non-Noetherian domains D.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [34]                       jesse.elliott@csuci.edu
Krull dimension

Rings of
Integer-Valued
Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                   1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [34]                       jesse.elliott@csuci.edu
Krull dimension

Rings of
Integer-Valued
Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems

UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
Krull dimension

Rings of
Integer-Valued
Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems

“dim Int” Conjecture
dim(Int(D)) ≤ dim(D[X]) for every domain D.

UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
Krull dimension

Rings of
Integer-Valued
Polynomials        Generally speaking, the Krull dimension of D[X] and Int(D) are
Jesse Elliott       hard to compute for non-Noetherian domains D. For D[X], it is
Introduction          known that
Prüfer domains

Integer-valued                     1 + dim(D) ≤ dim(D[X]) ≤ 1 + 2 dim(D).
polynomial rings

Krull domains
and PVMDs

Universal             It is not hard to show for all D that
properties of IVP
rings

Summmary of                              dim(Int(D)) ≥ dim(D[X]) − 1.
open problems

“dim Int” Conjecture
dim(Int(D)) ≤ dim(D[X]) for every domain D.

Equality holds for Noetherian domains and PVMD’s.

UCLA NTS 02/25/08                             [34]                    jesse.elliott@csuci.edu
A conjecture

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction

Prüfer domains        If D is a Krull domain, or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs             for every set X.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [35]                      jesse.elliott@csuci.edu
A conjecture

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction

Prüfer domains        If D is a Krull domain, or if D is a Prüfer domain such that
Integer-valued        Int(Dm ) = Int(D)m for every maximal ideal m of D, then the ho-
polynomial rings

Krull domains
momorphism θX : T ∈X Int(D) −→ Int(DX ) is an isomorphism
and PVMDs             for every set X.
Universal
properties of IVP
rings

Summmary of           Conjecture
open problems
If D is a PVMD such that Int(Dm ) = Int(D)m for every maximal
t-ideal m of D (which holds if D is a domain “of Krull-type”), then
the homomorphism θX is an isomorphism for every set X.

UCLA NTS 02/25/08                           [35]                        jesse.elliott@csuci.edu
A universal property of IVP rings

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Int(D) is not functorial in D. Nevertheless, it is possible to
characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
A universal property of IVP rings

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Int(D) is not functorial in D. Nevertheless, it is possible to
characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
A domain A containing D, say, with fraction ﬁeld L, is a
polynomial rings
polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
A universal property of IVP rings

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Int(D) is not functorial in D. Nevertheless, it is possible to
characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
A domain A containing D, say, with fraction ﬁeld L, is a
polynomial rings
polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings                 Proposition
Summmary of
open problems
For any inﬁnite domain D and any set X, the domain Int(DX )
is the free PC extension of D generated by X.

UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
A universal property of IVP rings

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Int(D) is not functorial in D. Nevertheless, it is possible to
characterize Int(D) uniquely with a universal property.
Introduction

Prüfer domains

Integer-valued
A domain A containing D, say, with fraction ﬁeld L, is a
polynomial rings
polynomially complete (PC) extension of D if any polynomial
Krull domains
and PVMDs             f (X) ∈ L[X] that maps D into A actually maps all of A into A.
Universal
properties of IVP
rings                 Proposition
Summmary of
open problems
For any inﬁnite domain D and any set X, the domain Int(DX )
is the free PC extension of D generated by X. In other words,
Int(DX ) is a PC extension of D, and for any PC extension A of D
and any map ϕ : X −→ A, there is a unique D-algebra homomor-
phism Int(DX ) −→ A sending T to ϕ(T ) for all T ∈ X.

UCLA NTS 02/25/08                            [36]                         jesse.elliott@csuci.edu
Example

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings      Example. A domain A containing Z is a PC extension of Z iff
a
Krull domains
and PVMDs
n ∈ A ⊗Z Q lies in A for every a ∈ A and every positive integer
n. Such a domain A is said to be a binomial domain.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [37]                       jesse.elliott@csuci.edu
Example

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings      Example. A domain A containing Z is a PC extension of Z iff
a
Krull domains
and PVMDs
n ∈ A ⊗Z Q lies in A for every a ∈ A and every positive integer
n. Such a domain A is said to be a binomial domain.
Universal
properties of IVP
rings

Summmary of
Int(ZX ) is the free binomial domain generated by X.
open problems

UCLA NTS 02/25/08                           [37]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
2  Int(D) ⊆ Int(A).
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
2  Int(D) ⊆ Int(A).
and PVMDs

Universal
3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [38]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
2  Int(D) ⊆ Int(A).
and PVMDs

Universal
3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings                   4  For every ideal m of D with ﬁnite residue ﬁeld, and for every
Summmary of
open problems
prime ideal P of A lying over m, one has mAP = PAP and
A/P = D/m.

UCLA NTS 02/25/08                           [38]                       jesse.elliott@csuci.edu
Over Dedekind domains

Rings of
Integer-Valued
Polynomials
Theorem (Gerboud, 1993)
Jesse Elliott

Introduction
For any extension A of an inﬁnite Dedekind domain D, the follow-
Prüfer domains
ing conditions are equivalent.
Integer-valued          1  A is a PC extension of D.
polynomial rings

Krull domains
2  Int(D) ⊆ Int(A).
and PVMDs

Universal
3  Int(A) is the A-module generated by Int(D).
properties of IVP
rings                   4  For every ideal m of D with ﬁnite residue ﬁeld, and for every
Summmary of
open problems
prime ideal P of A lying over m, one has mAP = PAP and
A/P = D/m.

Condition (4) can be rephrased as saying that A is unramiﬁed,
and has trivial residue ﬁeld extensions, at every maximal ideal
m of D with ﬁnite residue ﬁeld.

UCLA NTS 02/25/08                           [38]                       jesse.elliott@csuci.edu
Example

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
For example, Int(Z) is a polynomially complete extension of Z.
Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [39]                        jesse.elliott@csuci.edu
Example

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains
For example, Int(Z) is a polynomially complete extension of Z.
Integer-valued
polynomial rings      Therefore for every prime ideal P = mp,α of Int(Z) lying over a
Krull domains
and PVMDs             prime p in Z one has
Universal
properties of IVP
rings
PInt(Z)P = pInt(Z)P
Summmary of
open problems         and
Int(Z)/P = Z/pZ.

UCLA NTS 02/25/08                           [39]                        jesse.elliott@csuci.edu
Over Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem
Prüfer domains

Integer-valued        For any ﬂat extension A of an inﬁnite Krull domain D, the following
polynomial rings
conditions are equivalent.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
Over Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem
Prüfer domains

Integer-valued        For any ﬂat extension A of an inﬁnite Krull domain D, the following
polynomial rings
conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
Over Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem
Prüfer domains

Integer-valued        For any ﬂat extension A of an inﬁnite Krull domain D, the following
polynomial rings
conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
Over Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem
Prüfer domains

Integer-valued        For any ﬂat extension A of an inﬁnite Krull domain D, the following
polynomial rings
conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
3 Int(A) is the A-module generated by Int(D).
open problems

UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
Over Krull domains

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Theorem
Prüfer domains

Integer-valued        For any ﬂat extension A of an inﬁnite Krull domain D, the following
polynomial rings
conditions are equivalent.
Krull domains
and PVMDs               1 A is a PC extension of D.
Universal
properties of IVP       2 Int(D) ⊆ Int(A).
rings

Summmary of
3 Int(A) is the A-module generated by Int(D).
open problems
4 A is unramiﬁed, and has trivial residue ﬁeld extensions, at
every height one prime ideal of D with ﬁnite residue ﬁeld.

UCLA NTS 02/25/08                           [40]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
If (APC) holds, we will say that A is an almost polynomially
Summmary of
open problems         complete extension of D.

UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomiallly complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
If (APC) holds, we will say that A is an almost polynomially
Summmary of
open problems         complete extension of D.

If (WPC) holds, we will say that A is a weakly polynomially
complete extension of D.

UCLA NTS 02/25/08                            [41]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems

UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems

(PC) ⇔ (WPC) if D is an inﬁnite Krull domain and A is ﬂat as a
D-module.

UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
In general one has (PC) ⇒ (APC) ⇒ (WPC).
Summmary of
open problems

(PC) ⇔ (WPC) if D is an inﬁnite Krull domain and A is ﬂat as a
D-module.

(APC) ⇔ (WPC) if for all X the homomorphism
θX : T ∈X Int(D) −→ Int(DX ) is surjective.

UCLA NTS 02/25/08                            [42]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
The extension Z[T /2] of Z[T ] is APC but not PC.
Summmary of
open problems

UCLA NTS 02/25/08                            [43]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings
The extension Z[T /2] of Z[T ] is APC but not PC.
Summmary of
open problems

Indeed, Int(Z[T ]) = Z[T ][X] and every extension of Z[T ] is APC.
2
−X
However, the polynomial X 2 maps Z[T ] into Z[T /2] but does
not map all of Z[T /2] into itself, and therefore the extension
Z[T /2] of Z[T ] is not PC.

UCLA NTS 02/25/08                            [43]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott       Let A be a domain containing a domain D. Consider the following
Introduction
conditions.
Prüfer domains         1   (PC) A is a polynomially complete extension of D.
Integer-valued
polynomial rings       2   (APC) Int(Dn ) ⊆ Int(An ) for every positive integer n.
Krull domains
and PVMDs
3   (WPC) Int(D) ⊆ Int(A).
Universal
properties of IVP
rings

Summmary of
Problem
open problems
Does there exist an extension A of a some domain D that is WPC
but not APC?

UCLA NTS 02/25/08                            [44]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Proposition
Prüfer domains

Integer-valued        For any domain D and any set X, the domain Int(DX ) is the free
polynomial rings
APC extension of D generated by X.
Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [45]                      jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction
Proposition
Prüfer domains

Integer-valued        For any domain D and any set X, the domain Int(DX ) is the free
polynomial rings
APC extension of D generated by X.
Krull domains
and PVMDs

Universal
properties of IVP
rings
Problem
Summmary of
open problems
For which domains D is Int(DX ) a free WPC extension of D gen-
erated by X for every set X? Find an example, if one exists, of a
domain D for which this does not hold.

UCLA NTS 02/25/08                          [45]                       jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction
Let D be a domain. For any set X, let Intw (DX ) denote the inter-
Prüfer domains

Integer-valued
section of every subring of Int(DX ) containing D[X] that is closed
polynomial rings      under pre-composition by every element of Int(D). The domain
Krull domains
and PVMDs
Intw (DX ) is the free WPC extension of D generated by X.
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                           [46]                        jesse.elliott@csuci.edu
Other universal properties

Rings of
Integer-Valued
Polynomials

Jesse Elliott
Proposition
Introduction
Let D be a domain. For any set X, let Intw (DX ) denote the inter-
Prüfer domains

Integer-valued
section of every subring of Int(DX ) containing D[X] that is closed
polynomial rings      under pre-composition by every element of Int(D). The domain
Krull domains
and PVMDs
Intw (DX ) is the free WPC extension of D generated by X.
Universal
properties of IVP
rings

Summmary of
Proposition
open problems
Let D be a domain. Every WPC extension of D is APC iff
Int(DX ) = Intw (DX ) for every set X. Both of these conditions
hold if the homomorphism θX : T ∈X Int(D) −→ Int(DX ) is sur-
jective for every set X.

UCLA NTS 02/25/08                           [46]                        jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                [47]        jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X?
Prüfer domains

Integer-valued
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [47]                       jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
extension of a domain D APC?
polynomial rings

Krull domains
and PVMDs

Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                         [47]                       jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not ﬂat as a D-module?
Universal
properties of IVP
rings

Summmary of
open problems

UCLA NTS 02/25/08                          [47]                       jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not ﬂat as a D-module?
Universal
properties of IVP
3   Does there exist a domain D and a set X such that the
rings
natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?

UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not ﬂat as a D-module?
Universal
properties of IVP
3   Does there exist a domain D and a set X such that the
rings
natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?
4   Is dim(Int(D)) ≤ dim(D[X]) for every domain D?

UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
Summary of open problems

Rings of
Integer-Valued
Polynomials

Jesse Elliott
1   Is Int(DX ) the free WPC extension of D generated by X for
Introduction
any domain D and any set X? Equivalently, is every WPC
Prüfer domains

Integer-valued
extension of a domain D APC?
polynomial rings      2   Does there exist a domain D such that Int(D) is not free as a
Krull domains
and PVMDs                 D-module? not ﬂat as a D-module?
Universal
properties of IVP
3   Does there exist a domain D and a set X such that the
rings
natural homomorphism T ∈X Int(D) −→ Int(DX ) is not
Summmary of
open problems             injective? is not surjective? is neither injective nor surjective?
4   Is dim(Int(D)) ≤ dim(D[X]) for every domain D?
5   Does there exist a UFD or Krull domain D such that
dim D[X] > 1 + dim D?

UCLA NTS 02/25/08                            [47]                         jesse.elliott@csuci.edu
References

Rings of
Integer-Valued
Polynomials

Jesse Elliott

Introduction

Prüfer domains

Integer-valued        P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials,
polynomial rings
Mathematical Surveys and Monographs, V. 48, American
Krull domains
and PVMDs             Mathematical Society, 1997.
Universal
properties of IVP
rings                 J. Elliott, Universal properties of integer-valued polynomial rings,
Summmary of
open problems         Journal of Algebra 318 (2007) 68–92.

UCLA NTS 02/25/08                            [48]                         jesse.elliott@csuci.edu

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