# Let rt be a fuzzy singleton of R and A be a fuzzy module of R module

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```					J.Thi-Qar Sci.                         Vol.1 (1)                                 March/2008
ISSN 1991- 8690                                                      1661 - 0968 ‫الترقيم الدولي‬

On Quasi Prime Fuzzy Submodules and
Quasi Primary Fuzzy Submodules
Rabee Hadi Jari                      Hassan Kiream Hassan
Department of mathematics - College of Education,
University of Thi-Qar
thiqaruni.org.

Abstract
In this paper we give some results about fuzzy quasi prime
submodules, also,we study the notion of quasi primary fuzzy
0  State in 18O by using Core – Polarization submodules.
2                                          effects
Department of Physics, College of Science, Thi-qar University

1. Introduction:                              We record here some basic
The notion of prime fuzzy               concepts and clarify notions which are
submodules was introduced by Hatem        used in the next work .For details,we
Y.K [1] and some results a bout this      refer to [2,3].
concept are presented .In this paper      Definition 2.1 [2]:
concept and study the extension of        nonempty set M in to [0,1], then the
quasi primary fuzzy submodule by          set
fuzzifing the notion of (ordinary)
quasi primary fuzzy submodule and
give some characterizations about it      is called a level subset of X.
,also we give some properties related     Definition 2.2 [3]:
to it.
Let                   be a fuzzy set
Through out the paper R is a
commutative ring with unity and M is      in M, where, x  M , t  [0,1] , then xt is
an R-module.
2. Preliminaries:
J.Thi-Qar Sci.                                    Vol.1 (1)                                           March/2008

defined                    by                   :        Where A is a proper fuzzy submodule
t     if x  y                               of X.
xt ( y )  
o      if x  y for all y  M.
x t is called a fuzzy singleton.                           3. Quasi Prime Fuzzy submodules:
If x=0 and t=1, then:                                             In this section we turn our
attention to the extension of a quasi
Definition 2.3 [4]:                                         prime modules.
A fuzzy set X in R-module M is                         Definition 3.1 [1]:
called fuzzy R-module if and only if                              A fuzzy submodule A of a fuzzy
each x, y  M and r  R , then:                             module X of an R-module M is called
1 if y  0                                      quasi prime fuzzy submodule of X if
01 ( y)                                                   whenever         a s bk xt  A       for fuzzy
0 if y  0
Definition 2.3 [4]:                                         singleton             a s , bk of R and x t  X ,
A fuzzy set X in R-module M is                         implies that a s bk  A or b k xt  A .
called a fuzzy R-module if and only if                      Definition 3.2 [1]:
each x, y  M and r  R , then:                                   Let X be a fuzzy module of an R-
1. X ( x  y)  min X ( x), X ( y).                       module M. X is called a quasi prime
2. X (rx)  X ( x)                                          fuzzy module if and only if F-annA is
3. X(0)=1. (0 is the zero element of M).                    a prime fuzzy ideal of R for each
Definition 2.4 [5]:                                         nonempty fuzzy submodule A of X.
Let A and X be two fuzzy                            Equivalently,           F-ann(xt)=F-ann(rsxt)
module of an R-module M, A is called                        for each xt  X and for each fuzzy
a fuzzy submodule of X if and only if                       singleton rs of R such that
A X .                                                      r s  F  annX .
Definition 2.5 [5]:                                         Definition 3.3 [6]:
Let rt and xs be two fuzzy                                 Let A be a fuzzy submodule of a
singletons of R and M respectively.                         fuzzy module X, then A is called an
then rt x s  (rx )  , Where  =min{s,t}.                  essential       fuzzy          submodule       if
Definition 2.6 [5]:                                          A  B  01 ,for each nonempty fuzzy
Let rt be a fuzzy singleton of R                    submodule B of X.
and A be a fuzzy module of R-module.                        Lemma 3.4[6]:
Then                     for                     any             Let A be an essential fuzzy
w M ,                                                     submodule of X,if a k  X ,then there
sup{inf{t,A(x)}},if all w  rx for some x  M.
(rt A)(w)                                                 exists a fuzzy singleton rt of R such
o              otherwise
that 01  rt a k  A.
we noticed that if a fuzzy module X
Definition 2.7 [6]:
is quasi prime fuzzy module ,then any
Let X be a nonempty fuzzy
fuzzy submodule of X is a quasi prime
module of R-module M. The fuzzy
fuzzy module but the converse is not
annihilator of A denoted by F-annA is
true,see[1,Remarks and Examples
defined by:
2.1.3(4)]. However, we put certain
F  annA  (01 : A)  xt : xt A  01 , x  R, t  [0,1]
condition under which the converse is
true.
J.Thi-Qar Sci.                               Vol.1 (1)                                March/2008

Proposition 3.5:                                R if for each rt a k  I ,then either
Let A be an essential fuzzy                 rt  I or a k  I ,[7].
submodule of a fuzzy module X such
that      F-ann xt=F-ann rsxt for each          Corollary 3.7:
Let A be an essential fuzzy
xt  X and for a fuzzy singleton rs of
submodule of a fuzzy module X, if A is
R such that rs  F  annxt , then X is a        a faithful quasi prime module, then X
quasi prime fuzzy module if and only            is a faithful quasi prime fuzzy module.
if A is a quasi prime fuzzy module.             Proof:
Proof                                               Since A is a faithful, then F-
Suppose that A is a quasi prime            annA=01,            also, A  X ,   then
fuzzy R-module.                                  F  annX  F  annA by [6, Remark
To prove that 01 is a quasi prime fuzzy         1.2.18],         which      implies that
submodule of X.                                  F  annX  01 , but 01  F  annX ,
hence F-annX=01, That is X is faithful,
Let al bk xt  01                               moreover it follows that F-annX=F-
for fuzzy singletons al, bk of R and            annA.
xt  X .Since A is an essential fuzzy          Now, to prove X is a quasi prime fuzzy
submodule, thus by lemma 3.4, there             module,it is enough to prove that F-
exists a fuzzy singleton rs of R such           ann(xt)=F-ann(rsxt) for each xt  X
that             01  rs xt  A ,that      is   and for each fuzzy singleton rs of R
rs  F  annxt .                               such that r s  F  annX .
Hence al bk (rs xt )  01 . But A is a quasi       Let al  F  annrs xt for a fuzzy
prime fuzzy R-module, thus 01 is a              singleton rs of R and xt  X , then
quasi prime fuzzy submodule of A by              al (rs xt )  01 .
[1, proposition 3.4.1], so either
al (rs xt )  01 or bk (rs xt )  01 , thus
Consequently, (al rs ) xt  01 .       This
either            al  F  annrs xt       or    implies       that al rs  F  annxt ,   so
bk  F  annrs xt .                      But     al rs  F  annX . But F-annX=F-
F  annrs xt  F  annxt .
annA,then al rs  F  annA , thus F-
Consequently, either al  F  annxt or          annA is prime fuzzy ideal by [1,
bk  F  annxt , that is either al xt  01      definition 2.1.1], since rs  F  annA.
or bk xt  01 . Hence X is a quasi prime         So a l  F  annA , implies that
fuzzy module by [1, proposition 3.4.1].          al  F  annX , hence al X  01 , that
The converse follows directly by
is al xt  01 for all xt  X , hence
[1, Remarks and Examples 2.1.3(4)].
Definition 3.6 [6]:                              F  annrs xt  F  annxt …(1)
A fuzzy module X is called faithful       Now, let al  F  annxt for a fuzzy
if F-annX=01.                                   singleton al of R and xt  X .
Recall that a fuzzy ideal I of a           By lemma 3.4, there exists fuzzy
ring R is called a prime fuzzy ideal of         singleton rs of R such that
01  rs xt  A , that is rs  F  annxt ,
J.Thi-Qar Sci.                                     Vol.1 (1)                                          March/2008

so al rs xt   01 which implies that                            In this section we introduce the
al  F  annrs xt  ,            thus                     concept of quasi primary fuzzy
submodule by fuzzifying the ordinary
F  annxt  F  annrs xt   …(2)                           notion of primary submodule .
From (1) and (2) we get F-annxt=F-                         Definition 4.1 [5]:
annrsxt, thus X is a quasi prime fuzzy                           A fuzzy submodule A of a fuzzy
module by [1, theorem 2.1.9(3)].                           module X is called a primary fuzzy
Definition 3.8 [6]:                                        submodule if for each fuzzy singleton
A fuzzy module X is called fuzzy                     rt of R and a k  X such that
torsion free if F-annxt=01 for any
rt a k  A , then either
x t  X , x t  01              where
F  annxt  rk : rk fuzzy singleton of R, rk xt  01 
.
Lemma 3.9:                                                 ,where
Let R be an integral domain and                         ( A : X )  {rt : rt X  A, rt fuzzy sin gletonof R}
let X be a torsion free fuzzy module of                           Recall that if A is a fuzzy
R-module M, then X is a quasi prime                        submodule of a fuzzy module X of an
fuzzy module.                                              R-module and I is a fuzzy ideal of R
Proof:                                                     then (A:I) define as:
Suppose that X is a torsion free                        ( A : I )  {a k , a k I  A, a k  X } ,note
fuzzy module. To prove 01 is quasi
that :If I=(rt),then:
prime fuzzy submodule of X.
( A : (rt ))  {a k : rt a k  A, a k  X } ,[8].
Let al bk xt  01 ,for fuzzy singletons
Proposition 4.2:
al,bk of R and xt  X .                                          Let A be a proper fuzzy
If                                                         submodule of a fuzzy module X, then
bk xt  01 , then                                         the         following          statements       are
al  F  annbk xt ,                         since          equivalent:
1. A is a primary fuzzy submodule of
F  ann(bk xt )  F  anny ,        where                 X.
  min k, t and y=bx by [1, Remark                      2. (A:I) is a primary fuzzy submodule
1.2.14]      so     al  F  anny .   But                 of X for each fuzzy ideal I of R.
3. (A:(rt)) is a primary fuzzy
F  anny  01 because X is a torsion
submodule of X for each fuzzy
free, so a l  01 .                                        singleton rt of R.
On the other hand, 01  F  annX ,                         Proof:
which implies that al  F  annX ,                              1 2:
Suppose that rt x k  ( A : I ) for each
that is
x k  X and a fuzzy singleton rt of R
al X  01 , by [6, definition 1.2.17],                     and I be a fuzzy ideal of R.It follows
hence al xt  01 for all xt  X .                          that al rt x k  A for each fuzzy
Thus, X is a quasi prime fuzzy module                      singleton al of I, then either al x k  A
by [1,proposition 3.4.1].                                  for each fuzzy singleton al of I or
rt n  ( A : X ) , n  Z  .
4. Quasi Primary fuzzy submodule.
J.Thi-Qar Sci.                                       Vol.1 (1)                                           March/2008

So, either        x k  ( A : I ) or rtn X  A,             Now,       let       ak  A : rt B       for    all
since A  ( A : I ), by [8, theorem 3.3],                    rt B  A ,      hence     a k  ( A : rt B )
n
for
hence rt n X  ( A : I ) , implies that                     some n  Z       

and so a kn rt B  A ,which implies that
rt n  (( A : I ) : X ) ,it   follows,     either
a k rt  ( A : B ) . But (A:B) is a primary
n

x k  ( A : I ) or r  (( A : I ) : X ) , hence
n
t                                      fuzzy ideal of R and rt  ( A : B) ,so
a 
(A:I) is a primary fuzzy submodule of                          n m
X.                                                            k       ( A : B) for some m  Z  ,thus
2 3:                                                       a  ( A : B)
s
k                   where s=nm so that
It is clear.
3 1:                                                       ak  A : B ;                   that              is
It follows easily by taking r=1 and t=1.                      A : rt B  A : B              and         hence

Recall that a proper submodule N                         A : B  A : rt B .
ofan R-module M is said to be quasi
primary submodule if [N:K] is a
primary ideal of R if for each                               Conversely, to prove (A:B) is a
submodule K of M such that                                  primary uzzy ideal of R for each
N  K ,where                                                A  B.
[N:K]={r  R:rk  N},[9].                                   Suppose that a k rt  ( A : B) for a fuzzy
We shall fuzzify this concept as                       singleton ak and rt of R. If rt  ( A : B) ,
follows:
then ak  ( A : rt B)  A : rt B = A : B
Definition 4.3 :
A proper fuzzy submodule A of a                          and hence A is a quasi primary fuzzy
fuzzy module X is said to be quasi                          submodule of X.
primary fuzzy submodule if (A:B) is a                       Proposition 4.5:
primary fuzzy ideal of R for each                               Let A be a fuzzy submodule of a
fuzzy submodule B of X such that                            fuzzy module X, then A is a quasi
A B                          where                        primary submodule of X if and only if
(A:I) is a quasi primary fuzzy
( A : B)  rt / rt B  A, for a fuzzy singleton rt of R
submodule of X for each fuzzy ideal I
of R.
.                                                           Proof:
Proposition 4.4:                                                 Suppose that A is a quasi primary
Let A be a proper fuzzy                             fuzzy submodule of X.
submodule of a fuzzy module X, then                         By [8, theorem 3.3], (A:I) is a fuzzy
A is a quasi primary fuzzy submodule                        submodule of X.
of X if and only if A : B  A : rt B                            To prove (A:I) is a quasi primary
for each primary fuzzy submodule B                          fuzzy submodule of X, we must prove
of X and for each fuzzy singleton rt of                     ((A:I):B) is a primary fuzzy ideal for
R such that A  B and rt B  A .                            each ( A : I )  B.
Proof:                                                       A  ( A : I )  B by [8, lemma 2.16(1)] .
It is clear that A : B  A : rt B .                      Now, suppose that a k rt  (( A : I ) : B)
for fuzzy singleton rt and ak of R, then
J.Thi-Qar Sci.                                  Vol.1 (1)                                  March/2008

a k rt B  ( A : I ), so a k rt BI  A , suppose   Conversely, if 01 is quasi primary
that a k  (( A : I ) : B) , then a k BI  A ,     fuzzy submodule of X, to prove X is a
quasi primary fuzzy module. Since 01
hence a k rt I  ( A : B ) .                       is a quasi primary fuzzy submodule,
But, (A:B) is a primary fuzzy ideal by             then (01:A) is a primary fuzzy ideal for
definition 4.3 and a k I  ( A : B ) .Thus,        each nonempty fuzzy submodule.
rt n  ( A : B ) for some n  Z  and so           But (01:A)=F-annA, so X is a quasi
primary fuzzy R-module.
rt n B  A  ( A : I ) ,            therefore
rt  (( A : I ) : B ) and hence (A:I) is a
n
REFRENCES:
quasi primary fuzzy submodule of X.
[1] Hatem Y.K.,(2001), "Fuzzy Quasi –prime
The converse follows by taking                             Modules and Fuzzy Quasi –prime
I   R , where  R (a)  1 for all a  R                  submodules     ",   M.Sc.    thesis,
and by similar to the proof of [6,                         University of Baghdad, college of
Education Ibn- AL- Haithem ,(69)p.
proposition 2.1.16].
[2] Martines L.,(1996), "Fuzzy modules over
Recall that an R-module m is said                     fuzzy rings in connection with Fuzzy
to be a quasi primary module if annN                      ideal of Fuzzy ring ".       J.Fuzzy
is a primary ideal of R, for each non                     Mathematics, Vo1.4:843-857.
zero submodule N of M,[9].
[3]   Liu W. J,(1982),”Fuzzy invariant
Our attempt to fuzzify this concept                    subgroups and Fuzzy ideals”, fuzzy
is as follows:                                             sets                              and
systems,Vol.8:133-139.
Definition 4.6 :                                    [4] Nanda S. (1989),"Fuzzy Modules over
A fuzzy module X of R –module                          Fuzzy rings ", Bull. coll. Math. Soc.,
Vo1. 18 :197-200.
M is said to be quasi primary fuzzy                 [5] Mashinchi M. and Zahedi M.M.,(1992)"
module if and only if F-annA=(01:A) is                     On L-Fuzzy primary submodules ",
a primary fuzzy ideal for each                             Fuzzy sets and systems, Vol.49:231-
nonempty fuzzy submodule A of X .                          236.
[6]   Jari R.H.(2001)," prime         Fuzzy
submodules and prime           Fuzzy
Proposition 4.7:                                           modules ", M.Sc.              Thesis,
A fuzzy module X is a quasi                          University        of        Baghdad,
primary fuzzy module if and only if 01                     college of Education, Ibn –AL-
is a quasi primary fuzzy submodule of                      Haithem,(64)p.
X.
[7] Mukhrjie T.K.,(1989)” Prime Fuzzy
Proof:                                                   Ideals in Rings”, Fuzzy Sets and
Suppose that X is a quasi primary                    Systems           Vol. 32:337-341.
fuzzy module.                                       [8] Zahedi M.M.,(1992),"On L-Fuzzy
To prove 01 is a quasi primary fuzzy                     Residual                     Quotient
submodule of X, since X is a quasi                       Modules", Fuzzy sets              and
systems, Vol.51:333-344.
primary fuzzy submodule, then F-                    [9] Abdul-al-Kalik A.J.,(2005),”Primary
annA is a primary fuzzy ideal for each                   Modules”, M.Sc. thesis, University of
nonempty fuzzy submodule A of                            Baghdad,college      of     Education
X.Thus (01:A) is primary fuzzy ideal                     Ibn- AL-Haithem,(77)p.
and hence 01 is quasi primary fuzzy
submodule of X.
‫.‪J.Thi-Qar Sci‬‬                            ‫)1( 1.‪Vol‬‬                                   ‫8002/‪March‬‬

‫‪On Quasi Prime Fuzzy Submodules and Quasi Primary Fuzzy‬‬
‫‪Submodules‬‬
‫حسن كريم حسن‬                          ‫ربيع هادي جاري‬
‫قسم الرياضيات - كلية التربية - جامعة ذي قار‬

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