A New Type Fuzzy Module over Fuzzy Rings

Abstract

A new kind of fuzzy module over a fuzzy ring is introduced by generalizing Yuan and Lee's definition of the fuzzy group and Aktas and Çagman's definition of fuzzy ring. The concepts of fuzzy submodule, and fuzzy module homomorphism are studied and some of their basic properties are presented analogous of ordinary module theory.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Ece Yetkin 1 and Necati Olgun 2

Academic Editor:C. da Fonseca and Academic Editor:J. Hoff da Silva

1, Department of Mathematics, Faculty of Sciences and Arts, Marmara University, Istanbul, Turkey
2, Department of Mathematics, Faculty of Sciences and Arts, Gaziantep University, Gaziantep, Turkey

Received 27 August 2013; Accepted 15 January 2014; 5 March 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The concept of fuzzy subgroup of a group was first introduced by Rosenfeld [1] in 1971. Since then the theory of fuzzy algebra has been studied by many researchers [2-6]. In the definition of fuzzy subgroups, two types of fuzzy structures are observed in general. In the first type, the subset of a group G is fuzzy and the binary operation on G is nonfuzzy in the classical sense as Rosenfeld's definition [1]. In the second one, the set is nonfuzzy or classical and the binary operation is fuzzy in fuzzy sense as Yuan and Lee's [7] definition. By the use of Yuan and Lee's definition of fuzzy group based on fuzzy binary operation, Aktas and Çagman [8] defined a new kind of fuzzy ring.

In this study, we introduce a new kind of fuzzy module by using Yuan and Lee's definition of the fuzzy group and Aktas and Çagman's definition of fuzzy ring.

The fundamental properties of fuzzy groups and fuzzy rings are presented in Section 2. The concept of fuzzy module is introduced in Section 3. Finally, in Section 4, the concept of fuzzy submodule and fuzzy module homomorphism is presented and a fundamental homomorphism theorem of fuzzy module is obtained.

2. Preliminaries

In this section we will formulate the preliminary definitions and results that are required in this paper. Let θ ∈ [ 0,1 ) . Malik and Mordeson [4] gave the following definition.

Definition 1 (see [4]).

Let R and S be nonempty sets and let f be a fuzzy subset of R × S ; then f is called a fuzzy function R into S if

(1) ∀ x ∈ R , ∃ y ∈ S such that f ( x , y ) > θ ;

(2) ∀ x ∈ R , for all _{y 1} , _{y 2} ∈ S , f ( x , _{y 1} ) > θ and f ( x , _{y 2} ) > θ imply _{y 1} = _{y 2} .

By the use of Definition 1, Yuan and Lee [7] gave the following definition.

Definition 2 (see [7]).

Let G be a nonempty set and let R be a fuzzy subset of G × G × G . R is called a fuzzy binary operation on G if

(1) ∀ a , b ∈ G , ∃ c ∈ G such that R ( a , b , c ) > θ ;

(2) ∀ a , b , _{c 1} , _{c 2} ∈ G , R ( a , b , _{c 1} ) > θ and R ( a , b , _{c 2} ) > θ imply _{c 1} = _{c 2} .

Let R be a fuzzy binary operation on G ; then we have a mapping [figure omitted; refer to PDF] where F ( G ) = { A |" A : G [arrow right] [ 0,1 ] is a mapping } and [figure omitted; refer to PDF]

Let A = { a } and B = { b } and let R ( A , B ) be denoted as a [composite function] b ; then [figure omitted; refer to PDF]

Using the notations in (3), we have the following.

Definition 3 (see [7]).

Let G be a nonempty set and let R be a fuzzy binary operation on G . ( G , R ) is called a fuzzy group if the following conditions are true:

(G1) ∀ a , b , c , _{z 1} , _{z 2} ∈ G , ( ( a [composite function] b ) [composite function] c ) ( _{z 1} ) > θ and ( a [composite function] ( b [composite function] c ) ) ( _{z 2} ) > θ imply _{z 1} = _{z 2} ;

(G2) ∃ _{e o} ∈ G such that ( _{e o} [composite function] a ) ( a ) > θ and ( a [composite function] _{e o} ) ( a ) > θ for any a ∈ G ( _{e o} is called an identity element of G );

(G3) ∀ a ∈ G , ∃ b ∈ G such that ( a [composite function] b ) ( _{e o} ) > θ and ( b [composite function] a ) ( _{e o} ) > θ ( b is called an inverse element of a and denoted by ^{a - 1} ).

Proposition 4 (see [7]).

Consider ( ( a [composite function] b ) [composite function] c ) ( d ) > θ ... ( a [composite function] ( b [composite function] c ) ) ( d ) > θ .

Proposition 5 (see [7]).

H is a fuzzy subgroup of G if and only if

(1) ∀ a , b ∈ H , ∀ c ∈ G , ( a [composite function] b ) ( c ) > θ implies c ∈ H ;

(2) a ∈ H implies ^{a - 1} ∈ H .

Definition 6 (see [7]).

Let H be a fuzzy subgroup of G . Let [figure omitted; refer to PDF] and a H ( H a ) is called a left (right) coset of H .

Definition 7 (see [7]).

Let H be a fuzzy subgroup of G : [figure omitted; refer to PDF] and then H is called a normal fuzzy subgroup of G .

Definition 8 (see [8]).

Let ( G , R ) be a fuzzy subgroup. If [figure omitted; refer to PDF] then ( G , R ) is called abelian fuzzy group.

Theorem 9 (see [7]).

Let [ a H ] = { ^{a [variant prime]} H |" ^{a [variant prime]} H ~ a H } , a - = { ^{a [variant prime]} |" ^{a [variant prime]} ∈ G and ^{a [variant prime]} H ~ a H } , G / H = { [ a H ] |" a ∈ G } , and [figure omitted; refer to PDF] and then R - is a fuzzy binary relation on G / H .

Theorem 10 (see [7]).

( G / H , R - ) is a fuzzy group.

Definition 11 (see [7]).

Let ( _{G 1} , _{R 1} ) and ( _{G 2} , _{R 2} ) be two fuzzy groups and let f : _{G 1} [arrow right] _{G 2} be a mapping. If [figure omitted; refer to PDF] then f is called a fuzzy (group) homomorphism. If f is 1-1, it is called a fuzzy monomorphism. If f is onto, it is called a fuzzy epimorphism. If f is both 1-1 and onto, it is called a fuzzy isomorphism.

Let G be a fuzzy binary operation on R . Then we have a mapping [figure omitted; refer to PDF] where F ( R ) = { A |" A : R [arrow right] [ 0,1 ] is a mapping } and [figure omitted; refer to PDF] Let A = { a } and B = { b } and let G ( A , B ) and H ( A , B ) be denoted as a [composite function] b and a * b , respectively. Then [figure omitted; refer to PDF]

Using the notations of (11), we have the following.

Definition 12 (see [8]).

Let R be a nonempty set and let G and H be two fuzzy binary operations on R . Then ( R , G , H ) is called fuzzy ring if the following conditions hold:

(R1) ( R , G ) is an abelian fuzzy group;

(R2) ∀ a , b , c , _{z 1} , _{z 2} ∈ R , ( ( a * b ) * c ) ( _{z 1} ) > θ and ( a * ( b * c ) ) ( _{z 2} ) > θ imply _{z 1} = _{z 2} ;

(R3) ∀ a , b , c , _{z 1} , _{z 2} ∈ R , ( ( a [composite function] b ) * c ) ( _{z 1} ) > θ and ( ( a * c ) [composite function] ( b * c ) ) ( _{z 2} ) > θ imply _{z 1} = _{z 2} ; ( a * ( b [composite function] c ) ) ( _{z 1} ) > θ and ( ( a * b ) [composite function] ( a * c ) ) ( _{z 2} ) > θ imply _{z 1} = _{z 2} .

Definition 13 (see [8]).

Let ( R , G , H ) be a fuzzy ring.

(1) If ( a * b ) ( u ) > θ ... ( b * a ) ( u ) > θ , then ( R , G , H ) is said to be a commutative fuzzy ring.

(2) If ∃ _{e *} ∈ R such that ( a * _{e *} ) ( a ) > θ and ( _{e *} * a ) ( a ) > θ for every a ∈ R , then ( R , G , H ) is said to be fuzzy ring with identity.

(3) Let ( R , G , H ) be a fuzzy ring with identity. If ( a * b ) ( _{e *} ) > θ and ( b * a ) ( _{e *} ) > θ , ∀ a ∈ R , ∃ b ∈ R , then b is said to be an inverse element of a and is denoted by ^{a * - 1} .

Proposition 14 (see [8]).

Let ( R , G , H ) be a fuzzy ring and let S be a nonempty subset of R . Then ( S , G , H ) is a fuzzy subring of R if and only if

(1) ( a [composite function] b ) ( c ) > θ implies c ∈ S and ( a * b ) ( c ) > θ implies c ∈ S for all a , b ∈ S , c ∈ R ;

(2) a ∈ S implies ^{a - 1} ∈ S for all a ∈ S .

Definition 15 (see [8]).

A nonempty subset I of a fuzzy ring ( R , G , H ) is called a fuzzy ideal of R if the following conditions are satisfied.

(1) ∀ x , y ∈ I , ( x [composite function] y ) ( z ) > θ [implies] z ∈ I for all z ∈ R ;

(2) ∀ x ∈ I , ^{x - 1} ∈ I ;

(3) For all s ∈ I , for all r ∈ R , ( r * s ) ( x ) > θ [implies] x ∈ I and ( s * r ) ( y ) > θ [implies] y ∈ I , x , y ∈ R .

Definition 16 (see [8]).

Let I be a fuzzy ideal of fuzzy ring R and let Ω = { a [composite function] I |" a ∈ R } . One defines a relation over Ω : [figure omitted; refer to PDF]

3. Fuzzy Modules over Fuzzy Rings

Let ( R , G , H ) be a fuzzy ring and ( M , J ) be an abelian fuzzy group and let p be fuzzy function R × M into M . Then we have a mapping [figure omitted; refer to PDF] where F ( R ) = { A |" A : R [arrow right] [ 0,1 ] is a mapping } and F ( M ) = { N |" N : M [arrow right] [ 0,1 ] is a mapping } .

Let A = { r } and N = { m } , and let p ( A , N ) and J ( a , b ) be denoted as r [ecedil]9; m and a [ecedil]5; b , respectively. Then [figure omitted; refer to PDF]

Using the notations (14), we have the following.

Definition 17.

Let ( R , G , H ) be a fuzzy ring and let ( M , J ) be an abelian fuzzy group. M is called a (left) fuzzy module over R or (left) R -fmodule together with a fuzzy function p : R × M [arrow right] M if the following conditions hold. For all r , _{r 1} , _{r 2} ∈ R and for all m , _{m 1} , _{m 2} ∈ M ,

(M1) ( r [ecedil]9; ( _{m 1} [ecedil]5; _{m 2} ) ) ( x ) > θ and ( ( r [ecedil]9; _{m 1} ) [ecedil]5; ( r [ecedil]9; _{m 2} ) ) ( y ) > θ imply x = y ;

(M2) ( ( _{r 1} [composite function] _{r 2} ) [ecedil]9; m ) ( x ) > θ and ( ( _{r 1} [ecedil]9; m ) [ecedil]5; ( _{r 2} [ecedil]9; m ) ) ( y ) > θ imply x = y ;

(M3) ( ( _{r 1} * _{r 2} ) [ecedil]9; m ) ( x ) > θ and ( _{r 1} [ecedil]9; ( _{r 2} [ecedil]9; m ) ) ( y ) > θ imply x = y .

Proposition 18.

Let ( R , G , H ) be a fuzzy ring and let ( M , J ) be an R -fmodule; then for all r , _{r 1} , _{r 2} ∈ R , m , _{m 1} , _{m 2} ∈ M ,

(1) ( r [ecedil]9; ( _{m 1} [ecedil]5; _{m 2} ) ) ( x ) > θ ... ( ( r [ecedil]9; _{m 1} ) [ecedil]5; ( r [ecedil]9; _{m 2} ) ) ( x ) > θ ;

(2) ( ( _{r 1} [composite function] _{r 2} ) [ecedil]9; m ) ( x ) > θ ... ( ( _{r 1} [ecedil]9; m ) [ecedil]5; ( _{r 2} [ecedil]9; m ) ) ( x ) > θ ;

(3) ( ( _{r 1} * _{r 2} ) [ecedil]9; m ) ( x ) > θ ... ( _{r 1} [ecedil]9; ( _{r 2} [ecedil]9; m ) ) ( x ) > θ .

Proof.

(1) Let ( r [ecedil]9; ( _{m 1} [ecedil]5; _{m 2} ) ) ( x ) > θ and let _{x 1} , _{x 2} , y ∈ M such that p ( r , _{m 1} , _{x 1} ) > θ , p ( r , _{m 2} , _{x 2} ) > θ , and J ( _{x 1} , _{x 2} , y ) > θ . By [figure omitted; refer to PDF] we get x = y from (M1) and so ( ( r [ecedil]9; _{m 1} ) [ecedil]5; ( r [ecedil]9; _{m 2} ) ) ( x ) > θ . Similarly by ( ( r [ecedil]9; _{m 1} ) [ecedil]5; ( r [ecedil]9; _{m 2} ) ) ( x ) > θ we have ( r [ecedil]9; ( _{m 1} [ecedil]5; _{m 2} ) ) ( x ) > θ .

It is easy to prove (2) and (3) similar to the proof of (1).

Proposition 19.

Let ( R , G , H ) be a fuzzy ring with zero element _{e o} and ( M , J ) be a left R -fmodule with identity element _{e J} . Then for all r ∈ R , m ∈ M ,

(1) ( r [ecedil]9; _{e J} ) ( _{e J} ) > θ ;

(2) ( _{e o} [ecedil]9; m ) ( _{e J} ) > θ ;

(3) ( r [ecedil]9; m ) ( x ) > θ [implies] ( r [ecedil]9; ^{m - 1} ) ( ^{x - 1} ) > θ ;

(4) ( r [ecedil]9; m ) ( x ) > θ [implies] ( ^{r - 1} [ecedil]9; m ) ( ^{x - 1} ) > θ .

Proof.

(1) Let x ∈ M such that ( r [ecedil]9; _{e J} ) ( x ) > θ . Then [figure omitted; refer to PDF]

It follows that ( ( r [ecedil]9; _{e J} ) [ecedil]5; ( r [ecedil]9; _{e J} ) ) ( x ) > θ from Proposition 18. Then [figure omitted; refer to PDF]

Thus J ( x , x , x ) > θ and x = _{e J} from Proposition 2.1 in [7].

(2) Let x ∈ M such that ( _{e o} [ecedil]9; m ) ( x ) > θ . Then [figure omitted; refer to PDF]

It follows that ( ( _{e o} [ecedil]9; m ) [ecedil]5; ( _{e o} [ecedil]9; m ) ) ( x ) > θ from Proposition 18. Then [figure omitted; refer to PDF]

Thus similar to (1), J ( x , x , x ) > θ and so x = _{e J} .

(3) Let p ( r , m , x ) > θ and let y ∈ M such that p ( r , ^{m - 1} , y ) > θ . Since [figure omitted; refer to PDF] by Proposition 18 we have ( ( r [ecedil]9; m ) [ecedil]5; ( r [ecedil]9; ^{m - 1} ) ) ( _{e J} ) > θ . Hence [figure omitted; refer to PDF]

Therefore J ( x , y , _{e J} ) > θ and consequently y = ^{x - 1} .

(4) It is obtained similar to (3).

Proposition 20.

If ( R , G , H ) is a fuzzy ring and K is any fuzzy subring of R , then R is a left K -fmodule.

Proof.

Let ( R , G , H ) be a fuzzy ring and let ( K , G , H ) be a fuzzy subring of R . Consider the mapping [figure omitted; refer to PDF] defined by p ( k , r ) = H ( k , r ) . It is obviously a fuzzy function which satisfies the conditions in Definition 17. Moreover observe that ( R , G ) is necessarily an abelian fuzzy group. Consequently R is a left K -fmodule.

4. Fuzzy Submodule and Fuzzy Module Homomorphism

Definition 21.

Let ( R , G , H ) be a fuzzy ring, ( M , J ) an R -fmodule, and N a nonempty subset of M . If ( N , J ) is an R -fmodule, N is called a fuzzy submodule of M .

Proposition 22.

Let ( R , G , H ) be a fuzzy ring, ( M , J ) an R -fmodule, and N a nonempty subset of M . Then N is a fuzzy submodule of M if and only if

(1) ( N , J ) is a fuzzy subgroup of ( M , J ) ;

(2) for all r ∈ R , b ∈ N , ( r [ecedil]9; b ) ( c ) > θ implies c ∈ N .

Proposition 23.

If { _{N i} |" i ∈ I } is a family of fuzzy submodules of a fuzzy module M , then _{... i ∈ I N i} is a fuzzy submodule of M .

Definition 24.

Let A and B be two fuzzy modules over a fuzzy ring ( R , G , H ) with a function p : R × M [arrow right] M . A function f : A [arrow right] B is an R -fmodule homomorphism which provided that, for all a , b ∈ A and r ∈ R ,

(1) G ( a , b , x ) > θ implies G ( f ( a ) , f ( b ) , f ( x ) ) > θ ;

(2) p ( r , a , x ) > θ implies p ( r , f ( a ) , f ( x ) ) > θ .

Clearly, an R -fmodule homomorphism f : A [arrow right] B is necessarily an abelian fuzzy group homomorphism. Consequently the same terminology is used for fuzzy modules: f is an R -fmodule monomorphism (resp., epimorphism, isomorphism) if it is injective (resp., surjective, bijective) as fuzzy group homomorphisms.

Let f : A [arrow right] B be an R -fmodule homomorphism. Then the kernel and the image of f as fuzzy group homomorphisms are denoted by [figure omitted; refer to PDF] respectively.

Theorem 25.

Let ( R , G , H ) be a fuzzy ring and let f : A [arrow right] B be an R -fmodule homomorphism. Then

(1) f is an R -fmodule monomorphism if and only if Ker ... f = { _{e A} } ;

(2) f : A [arrow right] B is an R -fmodule isomorphism if and only if there exists a fuzzy module homomorphism g : B [arrow right] A such that g f = _{e A} and f g = _{e B} .

Proposition 26.

Let f : A [arrow right] B be an R -fmodule homomorphism. Then

(1) Ker ... f is a fuzzy submodule of A ;

(2) Im ... f is a fuzzy submodule of B ;

(3) if C is any fuzzy submodule of B , then ^{f - 1} ( C ) = { a ∈ A |" f ( a ) ∈ C } is a fuzzy submodule of A .

Proof.

(1) Ker ... f is a fuzzy subgroup of the abelian fuzzy group A from Theorem 26 in [8]. Let r ∈ R and a ∈ Ker ... f such that p ( r , a , x ) > θ . Since f is an R -fmodule homomorphism, p ( r , f ( a ) , f ( x ) ) > θ . On the other hand, as a ∈ Ker ... f we have f ( a ) = _{e B} . Therefore p ( r , _{e B} , f ( x ) ) > θ and so f ( x ) = _{e B} from Proposition 19. So x ∈ Ker ... f is obtained.

(2) Im ... f is a fuzzy subgroup of the abelian fuzzy group A from Theorem 26 in [8]. For any r ∈ R , b ∈ Im ... f , there exists a ∈ A such that b = f ( a ) . Let x ∈ A such that H ( r , a , x ) > θ . Since f is an R -fmodule homomorphism, H ( r , f ( a ) , f ( x ) ) > θ which means H ( r , b , f ( x ) ) > θ and so f ( x ) ∈ B .

(3) ^{f - 1} ( C ) is a fuzzy subgroup of the abelian fuzzy group A from Theorem 5.2 in [7]. Let r ∈ R and x ∈ ^{f - 1} ( C ) such that H ( r , x , u ) > θ . Since H ( r , f ( x ) , f ( u ) ) > θ and f ( x ) ∈ C , we have that f ( u ) ∈ C and u ∈ ^{f - 1} ( C ) . This completes the proof.

Proposition 27.

Let I be a left fuzzy ideal of a fuzzy ring ( R , G , H ) , ( A , J ) an R -fmodule, and a ∈ A . Consider the set B = I * a = { x ∈ R |" H ( r , a , x ) > θ , r ∈ I } . Then

(1) B is a fuzzy submodule of A ;

(2) The map [straight phi] : I [arrow right] B given by [straight phi] ( r ) = r * a is an R -fmodule epimorphism.

Proof.

(1) First we show that B is a fuzzy subgroup of A . Let x , y ∈ B ; then there exist _{r 1} , _{r 2} ∈ I such that H ( _{r 1} , a , x ) > θ and H ( _{r 2} , a , y ) > θ . From Proposition 19, H ( _{r 2} , a , y ) > θ implies H ( ^{r 2 - 1} , a , ^{y - 1} ) . Since I is a left fuzzy ideal, there exists r ∈ I such that G ( _{r 1} , ^{r 2 - 1} , r ) > θ .

Let u ∈ B such that H ( r , a , u ) > θ . Then [figure omitted; refer to PDF]

On the other hand, since H ( ^{r 2 - 1} , a , ^{y - 1} ) > θ and ( ( _{r 1} * a ) [ecedil]5; ( ^{r 2 - 1} * a ) ) ( u ) > θ from Proposition 18, we obtain [figure omitted; refer to PDF]

Thus J ( x , ^{y - 1} , u ) > θ , u ∈ B , which means that ( B , J ) is a fuzzy subgroup of ( A , J ) .

Now consider a mapping p : R × B [arrow right] B defined by p ( r , b ) = H ( r , b ) . Let ( r [ecedil]9; b ) ( c ) > θ , for any r ∈ R , b ∈ B . Since b ∈ B , there exists r ∈ I such that H ( r , a , b ) > θ .

Let s ∈ I such that H ( r , r , s ) > θ . Then [figure omitted; refer to PDF] and it follows that ( ( r * r ) [ecedil]9; a ) ( c ) > θ . Thus [figure omitted; refer to PDF] and we have H ( s , a , c ) > θ . Since s ∈ I , c ∈ B is obtained. Therefore B is a fuzzy submodule of A .

(2) Let r , _{r 1} , _{r 2} ∈ I such that G ( _{r 1} , _{r 2} , r ) > θ and let x , _{x 1} , _{x 2} ∈ B such that [straight phi] ( r ) = x , [straight phi] ( _{r 1} ) = _{x 1} , and [straight phi] ( _{r 2} ) = _{x 2} . So, H ( r , a , x ) > θ , H ( _{r 1} , a , _{x 1} ) > θ , and H ( _{r 2} , a , _{x 2} ) > θ . Since [figure omitted; refer to PDF] we have [figure omitted; refer to PDF]

It follows that J ( _{x 1} , _{x 2} , x ) > θ which means J ( [straight phi] ( _{r 1} ) , [straight phi] ( _{r 2} ) , [straight phi] ( r ) ) > θ .

Finally we show that if r ∈ R , k ∈ I such that ( r [ecedil]9; k ) ( s ) > θ , then ( r [ecedil]9; [straight phi] ( k ) ) ( [straight phi] ( s ) ) > θ . For this purpose, let [straight phi] ( k ) = x ∈ B and [straight phi] ( s ) = y ∈ B ; then H ( k , a , x ) > θ and H ( s , a , y ) > θ . Since [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] and it follows that H ( r , x , y ) > θ , and consequently we obtain ( r [ecedil]9; [straight phi] ( k ) ) ( [straight phi] ( s ) ) > θ . [straight phi] : I [arrow right] B is obviously surjective and [straight phi] is an R -fmodule epimorphism.

Proposition 28.

Let ( R , G , H ) be a fuzzy ring and B a fuzzy submodule of an R -fmodule ( A , J ) and let _{a 1} [ecedil]5; B ~ _{a 2} [ecedil]5; B , a [ecedil]5; B ~ ^{a [variant prime]} [ecedil]5; B , and c [ecedil]5; B ~ ^{c [variant prime]} [ecedil]5; B . Consider a mapping defined by [figure omitted; refer to PDF]

If p ( r , _{a 1} , x ) > θ and p ( r , _{a 2} , y ) > θ , then x [ecedil]5; B ~ y [ecedil]5; B .

Proof.

Let p ( r , _{a 1} , x ) > θ and p ( r , _{a 2} , y ) > θ . Since _{a 1} [ecedil]5; B ~ _{a 2} [ecedil]5; B and by Definition 16, there exists u ∈ B such that J ( ^{a 1 - 1} , _{a 2} , u ) > θ . Let a ∈ A and v ∈ B such that J ( _{a 1} , _{a 2} , a ) > θ and p ( r , u , v ) > θ . As B is an R -fmodule, v ∈ B . Then [figure omitted; refer to PDF] Let ( ( r [ecedil]9; ^{a 1 - 1} ) [ecedil]5; ( r [ecedil]9; _{a 2} ) ) ( w ) > θ . By Proposition 19, p ( r , _{a 1} , x ) > θ implies p ( r , ^{a 1 - 1} , ^{x - 1} ) > θ . Hence [figure omitted; refer to PDF] Thus w = v , so J ( ^{x - 1} , y , v ) > θ and consequently x [ecedil]5; B ~ y [ecedil]5; B .

Then we have the following result.

Theorem 29.

Let ( R , G , H ) be a fuzzy ring and B a fuzzy submodule of an R -fmodule ( A , J ) . Then A / B is an R -fmodule with the mapping p ¯ defined in Proposition 28.

Proof.

Since ( A , J ) is an abelian fuzzy group, ( B , J ) is necessarily a normal fuzzy subgroup of A . Hence the abelian factor fuzzy group A / B is well-defined. Since A is an R -fmodule, for all r ∈ R , _{a 1} ∈ A , there exists _{a 2} ∈ A such that p ( r , _{a 1} , _{a 2} ) > θ . Then p ¯ ( r [ _{a 1} [ecedil]5; B ] , [ _{a 2} [ecedil]5; B ] ) > p ( r , _{a 1} , _{a 2} ) > θ .

Let ( r [ecedil]9; [ a [ecedil]5; B ] ) ( [ x [ecedil]5; B ] ) > θ and ( r [ecedil]9; [ a [ecedil]5; B ] ) ( [ y [ecedil]5; B ] ) > θ . Then there exist _{a 1} , _{a 2} ∈ a ¯ , ^{x [variant prime]} ∈ x ¯ , and ^{y [variant prime]} ∈ y ¯ such that p ( r , _{a 1} , ^{x [variant prime]} ) > θ and p ( r , _{a 2} , ^{y [variant prime]} ) > θ . Since _{a 1} [ecedil]5; B ~ _{a 2} [ecedil]5; B , we have ^{x [variant prime]} [ecedil]5; B ~ ^{y [variant prime]} [ecedil]5; B from Proposition 28 and consequently [ x [ecedil]5; B ] = [ y [ecedil]5; B ] .

(1) Let ( r [ecedil]9; ( [ _{a 1} [ecedil]5; B ] [ecedil]5; [ _{a 2} [ecedil]5; B ] ) ) ( [ x [ecedil]5; B ] ) > θ and ( ( r [ecedil]9; _{[ a 1} [ecedil]5; B ] ) [ecedil]5; ( r [ecedil]9; [ _{a 2} [ecedil]5; B ] ) ) ( [ y [ecedil]5; B ] ) > θ and let a ∈ A such that J ( _{a 1} , _{a 2} , a ) > θ . Then there exist ^{a 1 [variant prime]} ∈ _{a 1} ¯ , ^{a 2 [variant prime]} ∈ _{a 2} ¯ , ^{x [variant prime]} , ^{x 1 [variant prime]} , ^{x 2 [variant prime]} ∈ A , ^{y [variant prime]} ∈ y ¯ , and _{b 1} , _{b 2} ∈ B such that [figure omitted; refer to PDF]

Let w ∈ A such that J ( ^{a 1 [variant prime]} , ^{a 2 [variant prime]} , w ) > θ . Then by J ( ^{a 1 [variant prime]} , _{b 1} , _{a 1} ) > θ , J ( ^{a 2 [variant prime]} , _{b 2} , _{a 2} ) > θ , J ( _{a 1} , _{a 2} , a ) > θ , and the proof of Theorem 4.2 in [7], we get b ∈ B such that J ( w , b , a ) > θ . Then J ( a , ^{b - 1} , w ) > θ . Since [figure omitted; refer to PDF] and A is an R -fmodule, we have ( r [ecedil]9; ( ^{a 1 [variant prime]} [ecedil]5; ^{a 2 [variant prime]} ) ) ( ^{y [variant prime]} ) > θ . So [figure omitted; refer to PDF] Hence p ( r , w , ^{y [variant prime]} ) > θ . Then [figure omitted; refer to PDF]

Let _{x 3} ∈ A , _{b 3} ∈ B such that p ( r , a , _{x 3} ) > θ and p ( r , ^{b - 1} , _{b 3} ) > θ . Then [figure omitted; refer to PDF]

It follows that J ( _{x 3} , _{b 3} , ^{y [variant prime]} ) > θ and so _{x 3} [ecedil]5; B ~ ^{y [variant prime]} [ecedil]5; B . Therefore we obtain ^{x [variant prime]} [ecedil]5; B ~ ^{y [variant prime]} [ecedil]5; B and consequently [ x [ecedil]5; B ] = [ y [ecedil]5; B ] .

(2) Let ( ( _{r 1} [composite function] _{r 2} ) [ecedil]9; [ a [ecedil]5; B ] ) ( [ x [ecedil]5; B ] ) > θ and ( ( _{r 1} [ecedil]9; [ a [ecedil]5; B ] ) [ecedil]5; ( _{r 2} [ecedil]9; [ a [ecedil]5; B ] ) ) ( [ y [ecedil]5; B ] ) > θ and let r ∈ R such that G ( _{r 1} , _{r 2} , r ) > θ . Then we have _{a 1} , _{a 2} , _{a 3} ∈ a ¯ , _{x 1} ∈ x ¯ , _{y 1} ∈ y ¯ , _{x 2} , _{x 3} ∈ A , and _{b 1} , _{b 2} ∈ B such that [figure omitted; refer to PDF]

Then, by [figure omitted; refer to PDF] we have ( ( _{r 1} [ecedil]9; _{a 1} ) [ecedil]5; ( _{r 1} [ecedil]9; _{b 1} ) ) ( _{x 2} ) > θ . So there exist _{u 1} ∈ A , _{v 1} ∈ B such that [figure omitted; refer to PDF]

Thus J ( _{u 1} , _{v 1} , _{x 2} ) > θ and so _{u 1} [ecedil]5; B ~ _{x 2} [ecedil]5; B .

Since [figure omitted; refer to PDF] we have ( ( _{r 2} [ecedil]9; _{a 1} ) [ecedil]5; ( _{r 2} [ecedil]9; _{b 2} ) ) ( _{x 3} ) > θ . So there exist _{u 2} ∈ A , _{v 2} ∈ B such that [figure omitted; refer to PDF]

Similarly, we get J ( _{u 2} , _{v 2} , _{x 3} ) > θ and so _{u 2} [ecedil]5; B ~ _{x 3} [ecedil]5; B .

Since [figure omitted; refer to PDF] we get ( ( _{r 1} [ecedil]9; _{a 1} ) [ecedil]5; ( _{r 2} [ecedil]9; _{a 1} ) ) ( _{x 1} ) > θ . Hence [figure omitted; refer to PDF]

Therefore, since _{u 1} [ecedil]5; B ~ _{x 2} [ecedil]5; B , _{u 2} [ecedil]5; B ~ _{x 3} [ecedil]5; B , J ( _{x 2} , _{x 3} , _{y 1} ) > θ , and J ( _{u 1} , _{u 2} , _{x 1} ) > θ , we obtain _{x 1} [ecedil]5; B ~ _{y 1} [ecedil]5; B and [ x [ecedil]5; B ] = [ y [ecedil]5; B ] .

(3) Finally, let ( ( _{r 1} * _{r 2} ) [ecedil]9; [ a [ecedil]5; B ] ) ( [ x [ecedil]5; B ] ) > θ and ( _{r 1} [ecedil]9; ( _{r 2} [ecedil]9; [ a [ecedil]5; B ] ) ) ( [ y [ecedil]5; B ] ) > θ and let r ∈ R such that H ( _{r 1} , _{r 2} , r ) > θ . Then there exist _{a 1} , _{a 2} ∈ a ¯ , _{x 1} ∈ x ¯ , _{y 1} ∈ y ¯ , _{x 2} , _{x 3} ∈ A , and _{b 1} , _{b 2} ∈ B such that [figure omitted; refer to PDF] and let w ∈ A such that p ( r , _{a 2} , w ) > θ . By [figure omitted; refer to PDF] we get w = _{y 1} and p ( r , _{a 2} , _{y 1} ) > θ .

Since _{a 1} [ecedil]5; B ~ _{a 2} [ecedil]5; B , p ( r , _{a 1} , _{x 1} ) > θ , and p ( r , _{a 2} , _{y 1} ) > θ , then we have _{x 1} [ecedil]5; B ~ _{y 1} [ecedil]5; B from Proposition 28 and consequently [ x [ecedil]5; B ] = [ y [ecedil]5; B ] .

Theorem 30 (fundamental homomorphism theorem of fuzzy modules).

Let ( R , G , H ) be fuzzy ring and f : A [arrow right] B an R -fmodule epimorphism. Then A / K is isomorphic to B where K = Ker ... f .

Proof.

We have shown that K = Ker ... f is a fuzzy submodule of A in Proposition 26. Hence A / K is an R -fmodule from Theorem 29. Now we define a mapping by [figure omitted; refer to PDF]

f - is well-defined, one to one and surjective fuzzy group homomorphism from Theorem 5.3 in [7]. Then it suffices to prove that, for any r ∈ R , a ∈ A , p ¯ ( r [ a [ecedil]5; K ] , [ b [ecedil]5; K ] ) > θ implies p ¯ ( r , f - ( [ a [ecedil]5; K ] ) , f - ( [ b [ecedil]5; K ] ) ) > θ . Let p ¯ ( r [ a [ecedil]5; K ] , [ b [ecedil]5; K ] ) > θ and let u ∈ B such that p ¯ ( r , f - ( [ a [ecedil]5; K ] ) , u ) > θ . Then we have ^{a [variant prime]} ∈ a ¯ , ^{b [variant prime]} ∈ b ¯ such that p ( r , ^{a [variant prime]} , ^{b [variant prime]} ) > θ .

Let w ∈ A such that p ( r , a , w ) > θ . Since f is an R -fmodule homomorphism, we get p ( r , f ( a ) , f ( w ) ) > θ , so u = f ( w ) . By ^{a [variant prime]} [ecedil]5; K ~ a [ecedil]5; K , p ( r , ^{a [variant prime]} , ^{b [variant prime]} ) > θ , p ( r , a , w ) > θ , and Proposition 28, we have ^{b [variant prime]} [ecedil]5; K ~ w [ecedil]5; K and so [ b [ecedil]5; K ] = [ w [ecedil]5; K ] . Therefore f ( w ) = f - ( [ w [ecedil]5; K ] ) = f - ( [ b [ecedil]5; K ] ) = f ( b ) and consequently u = f ( b ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] A. Rosenfeld, "Fuzzy groups," Journal of Mathematical Analysis and Applications , vol. 35, no. 3, pp. 512-517, 1971.

[2] V. N. Dixit, R. Kumar, N. Ajmal, "On fuzzy rings," Fuzzy Sets and Systems , vol. 49, no. 2, pp. 205-213, 1992.

[3] W.-J. Liu, "Fuzzy invariant subgroups and fuzzy ideals," Fuzzy Sets and Systems , vol. 8, no. 2, pp. 133-139, 1982.

[4] D. S. Malik, J. N. Mordeson, "Fuzzy homomorphisms of rings," Fuzzy Sets and Systems , vol. 46, no. 1, pp. 139-146, 1992.

[5] J. N. Mordeson, D. S. Malik Fuzzy Commutative Algebra , of Pure Mathematics Series, World Scientific, Singapore, 1998.

[6] T. K. Mukherjee, M. K. Sen, "On fuzzy ideals of a ring I," Fuzzy Sets and Systems , vol. 21, no. 1, pp. 99-104, 1987.

[7] X. Yuan, E. S. Lee, "Fuzzy group based on fuzzy binary operation," Computers & Mathematics with Applications , vol. 47, no. 4-5, pp. 631-641, 2004.

[8] H. Aktas, N. A. Çagman, "A type of fuzzy ring," Archive for Mathematical Logic , vol. 46, no. 3-4, pp. 165-177, 2007.