#### 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 Aktaş and Çağman’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.

#### 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 is fuzzy and the binary operation on 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, Aktaş and Çağman [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 Aktaş and Çağman’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 . Malik and Mordeson [4] gave the following definition.

*Definition 1 (see [4]). *Let and be nonempty sets and let be a fuzzy subset of ; then is called a fuzzy function into if(1), such that ;(2), for all , and imply .

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

*Definition 2 (see [7]). *Let be a nonempty set and let be a fuzzy subset of . is called a fuzzy binary operation on if (1), such that ;(2), and imply .

Let be a fuzzy binary operation on ; then we have a mapping
where and

Let and and let be denoted as ; then

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

*Definition 3 (see [7]). *Let be a nonempty set and let be a fuzzy binary operation on . is called a fuzzy group if the following conditions are true: (G1), and imply ;(G2) such that and for any ( is called an identity element of );(G3), such that and ( is called an inverse element of and denoted by ).

Proposition 4 (see [7]). *Consider .*

Proposition 5 (see [7]). * is a fuzzy subgroup of if and only if *(1), , *implies* ;(2) *implies* .

*Definition 6 (see [7]). *Let be a fuzzy subgroup of . Let
and is called a left (right) coset of .

*Definition 7 (see [7]). *Let be a fuzzy subgroup of :
and then is called a normal fuzzy subgroup of .

*Definition 8 (see [8]). *Let be a fuzzy subgroup. If
then is called abelian fuzzy group.

Theorem 9 (see [7]). *Let , and , , and
**
and then is a fuzzy binary relation on .*

Theorem 10 (see [7]). * is a fuzzy group.*

*Definition 11 (see [7]). *Let and be two fuzzy groups and let be a mapping. If
then is called a fuzzy (group) homomorphism. If is 1-1, it is called a fuzzy monomorphism. If is onto, it is called a fuzzy epimorphism. If is both 1-1 and onto, it is called a fuzzy isomorphism.

Let be a fuzzy binary operation on . Then we have a mapping
where and
Let and and let and be denoted as and , respectively. Then

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

*Definition 12 (see [8]). *Let be a nonempty set and let and be two fuzzy binary operations on . Then is called fuzzy ring if the following conditions hold: (R1) is an abelian fuzzy group;(R2), and imply ;(R3), and imply ; and imply .

*Definition 13 (see [8]). *Let be a fuzzy ring. (1)If , then is said to be a commutative fuzzy ring.(2)If such that and for every , then is said to be fuzzy ring with identity.(3)Let be a fuzzy ring with identity. If and , , , then is said to be an inverse element of and is denoted by .

Proposition 14 (see [8]). *Let be a fuzzy ring and let be a nonempty subset of . Then is a fuzzy subring of if and only if*(1) *implies* *and* *implies* *for all* ;(2) *implies* *for all* .

*Definition 15 (see [8]). *A nonempty subset of a fuzzy ring is called a fuzzy ideal of if the following conditions are satisfied. (1), for all ;(2), ;(3)For all , for all , and , .

*Definition 16 (see [8]). *Let be a fuzzy ideal of fuzzy ring and let . One defines a relation over :

#### 3. Fuzzy Modules over Fuzzy Rings

Let be a fuzzy ring and be an abelian fuzzy group and let be fuzzy function into . Then we have a mapping where and .

Let and , and let and be denoted as and , respectively. Then

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

*Definition 17. *Let be a fuzzy ring and let be an abelian fuzzy group. is called a (left) fuzzy module over or (left) -fmodule together with a fuzzy function if the following conditions hold. For all and for all ,(M1) and imply ;(M2) and imply ;(M3) and imply .

Proposition 18. *Let be a fuzzy ring and let be an -fmodule; then for all , ,*(1);(2);(3).

* Proof. *(1) Let and let such that , , and . By
we get from (M1) and so . Similarly by we have .

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

Proposition 19. *Let be a fuzzy ring with zero element and be a left -fmodule with identity element . Then for all , ,*(1);(2);(3);(4).

*Proof. *(1) Let such that . Then

It follows that from Proposition 18. Then

Thus and from Proposition 2.1 in [7].

(2) Let such that . Then

It follows that from Proposition 18. Then

Thus similar to (1), and so .

(3) Let and let such that . Since
by Proposition 18 we have . Hence

Therefore and consequently .

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

Proposition 20. *If is a fuzzy ring and is any fuzzy subring of , then is a left -fmodule.*

* Proof. *Let be a fuzzy ring and let ( be a fuzzy subring of . Consider the mapping
defined by . It is obviously a fuzzy function which satisfies the conditions in Definition 17. Moreover observe that is necessarily an abelian fuzzy group. Consequently is a left -fmodule.

#### 4. Fuzzy Submodule and Fuzzy Module Homomorphism

*Definition 21. *Let be a fuzzy ring, an -fmodule, and a nonempty subset of . If is an -fmodule, is called a fuzzy submodule of .

Proposition 22. *Let be a fuzzy ring, an -fmodule, and a nonempty subset of . Then is a fuzzy submodule of if and only if *(1) *is a fuzzy subgroup of* ;(2)*for all* , , *implies* .

Proposition 23. *If is a family of fuzzy submodules of a fuzzy module , then is a fuzzy submodule of .*

*Definition 24. *Let and be two fuzzy modules over a fuzzy ring with a function . A function is an -fmodule homomorphism which provided that, for all and ,(1) implies ;(2) implies .

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

Let be an -fmodule homomorphism. Then the kernel and the image of as fuzzy group homomorphisms are denoted by
respectively.

Theorem 25. *Let be a fuzzy ring and let be an -fmodule homomorphism. Then*(1)* is an -fmodule monomorphism if and only if ;*(2)* is an -fmodule isomorphism if and only if there exists a fuzzy module homomorphism such that and .*

Proposition 26. *Let be an -fmodule homomorphism. Then *(1) *is a fuzzy submodule of* ;(2) *is a fuzzy submodule of* ;(3)*if* *is any fuzzy submodule of* , *then* *is a fuzzy submodule of* .

* Proof. *(1) is a fuzzy subgroup of the abelian fuzzy group from Theorem 26 in [8]. Let and such that . Since is an -fmodule homomorphism, . On the other hand, as we have . Therefore and so from Proposition 19. So is obtained.

(2) is a fuzzy subgroup of the abelian fuzzy group from Theorem 26 in [8]. For any , , there exists such that . Let such that . Since is an -fmodule homomorphism, which means and so .

(3) is a fuzzy subgroup of the abelian fuzzy group from Theorem 5.2 in [7]. Let and such that . Since and , we have that and . This completes the proof.

Proposition 27. *Let be a left fuzzy ideal of a fuzzy ring , an -fmodule, and . Consider the set . Then *(1) *is a fuzzy submodule of* ;(2)*The map* *given by* *is an* *-fmodule epimorphism.*

*Proof. *(1) First we show that is a fuzzy subgroup of . Let ; then there exist such that and . From Proposition 19, implies . Since is a left fuzzy ideal, there exists such that .

Let such that . Then

On the other hand, since and from Proposition 18, we obtain

Thus , , which means that is a fuzzy subgroup of .

Now consider a mapping defined by . Let , for any , . Since , there exists such that .

Let such that . Then
and it follows that . Thus
and we have . Since , is obtained. Therefore is a fuzzy submodule of .

(2) Let such that and let such that , , and. So, , , and . Since
we have

It follows that which means .

Finally we show that if , such that , then . For this purpose, let and ; then and . Since
we have
and it follows that , and consequently we obtain . is obviously surjective and is an -fmodule epimorphism.

Proposition 28. *Let be a fuzzy ring and a fuzzy submodule of an -fmodule and let , , and . Consider a mapping defined by
**If and , then .*

*Proof. *Let and . Since and by Definition 16, there exists such that . Let and such that and . As is an -fmodule, . Then
Let . By Proposition 19, implies . Hence
Thus , so and consequently .

Then we have the following result.

Theorem 29. *Let be a fuzzy ring and a fuzzy submodule of an -fmodule . Then is an -fmodule with the mapping defined in Proposition 28.*

*Proof. *Since is an abelian fuzzy group, is necessarily a normal fuzzy subgroup of . Hence the abelian factor fuzzy group is well-defined. Since is an -fmodule, for all , , there exists such that . Then .

Let and . Then there exist , , and such that and . Since , we have from Proposition 28 and consequently .

(1) Let and and let such that . Then there exist , , , , and such that

Let such that . Then by , , , and the proof of Theorem 4.2 in [7], we get such that . Then . Since
and is an -fmodule, we have . So
Hence . Then

Let , such that and . Then

It follows that and so . Therefore we obtain and consequently .

(2) Let and and let such that . Then we have , , , , and such that

Then, by
we have . So there exist , such that

Thus and so .

Since
we have . So there exist , such that

Similarly, we get and so .

Since
we get . Hence

Therefore, since , , , and , we obtain and .

(3) Finally, let and and let such that . Then there exist , , , , and such that
and let such that . By
we get and .

Since , , and , then we have from Proposition 28 and consequently .

Theorem 30 (fundamental homomorphism theorem of fuzzy modules). *Let be fuzzy ring and an -fmodule epimorphism. Then is isomorphic to where .*

*Proof. *We have shown that is a fuzzy submodule of in Proposition 26. Hence is an -fmodule from Theorem 29. Now we define a mapping by

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 , , implies . Let and let such that . Then we have such that .

Let such that . Since is an -fmodule homomorphism, we get , so . By , , , and Proposition 28, we have and so . Therefore and consequently .

#### Conflict of Interests

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