`International Journal of Mathematics and Mathematical SciencesVolume 2011, Article ID 980320, 12 pageshttp://dx.doi.org/10.1155/2011/980320`
Research Article

## Fuzzy Small Submodule and Jacobson -Radical

1Department of Mathematics, Rajiv Gandhi University, Itanagar 791112, India
2Department of Mathematics, Gauhati University, Guwahati 781014, India

Received 22 December 2010; Accepted 2 March 2011

Academic Editor: Enrico Obrecht

Copyright © 2011 Saifur Rahman and Helen K. Saikia. 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.

#### Abstract

Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and sufficient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions. Finally, a relation between small -submodules and Jacobson -radical is established.

#### 1. Introduction

After the introduction of fuzzy sets by Zadeh [1], a number of applications of this fundamental concept have come up. Rosenfeld [2] was the first one to define the concept of fuzzy subgroups of a group. Since then many generalizations of this fundamental concept have been done in the last three decades. Naegoita and Ralescu [3] applied this concept to modules and defined fuzzy submodules of a module. Consequently, fuzzy finitely generated submodules, fuzzy quotient modules [4], radical of fuzzy submodules, and primary fuzzy submodules [5, 6] were investigated. Saikia and Kalita [7] defined fuzzy essential submodules and investigated various characteristics of such submodules. These modules play a prominent role in fuzzy Goldie dimension of modules.

In this paper we fuzzify various properties of small (or superfluous) submodules of a module. We define fuzzy small ephimorphism and fuzzy coessential extension of a fuzzy submodule. We investigate various characteristics of fuzzy small submodules. Necessary and sufficient conditions for fuzzy small submodules are established. We also investigate the nature of fuzzy small submodule under fuzzy direct sum. A relation regarding fuzzy small submodule of a module and fuzzy quotient module is also established. We attempt to fuzzify the well-known relation between the Jacobson radical and the small submodules of a module. In [8] Basnet et. al. have shown that the relation between the Jacobson radical and the small submodules of a module does not hold in fuzzy setting whereas we have tried to achieve the relation. It is established that the Jacobson -radical is the sum of all the small -submodules of a module. In case of a finitely generated module, the Jacobson -radical is also a small -submodule of the module under the condition that - possesses a maximal element.

#### 2. Basic Definitions and Notations

By we mean a commutative ring with unity 1 and denotes a -module. The zero elements of and are 0 and , respectively. A complete Heyting algebra is a complete lattice such that for all and for all , and .

Definition 2.1. A submodule of a module over a ring is said to be a small submodule of if for every submodule of with implies .
The notation indicates that is a small submodule of .

Fuzzy set on a nonempty set was introduced by Zadeh [1] in 1965. It is defined as follows.

Definition 2.2. By a fuzzy set of a module , we mean any mapping from to . By we will denote the set of all fuzzy subsets of . If is a mapping from to , where is a complete Heyting algebra then is called an -subset of . By we will denote the set of all -subsets of .
For each fuzzy set in and any , we define two sets , , which are called an upper level cut and a lower level cut of , respectively. The complement of , denoted by , is the fuzzy set on defined by . The support of a fuzzy set , denoted by , is a subset of defined by . The subset of is defined as .

Definition 2.3 (see [9]). If and then is defined as,

If = then is often called a fuzzy point and is denoted by . When then is known as the characteristic function of . From now onwards, we will denote the characteristic function of as .

If then(1), if and only if ;(2);(3).

For any family of fuzzy subsets of , where is any nonempty index set,(4);(5) for all ;(6).

Definition 2.4 (see [9]). Let and be any two nonempty sets, and be a mapping. Let and then the image and the inverse image are defined as follows: for all and for all .

Definition 2.5 (see [9]). Let and . Then is a fuzzy subset of and it is defined by for all .

Definition 2.6 (see [9]). A fuzzy set of is called a fuzzy ideal, if it satisfies the following properties:(1),(2), for all .

The following definition is given by Naegoita and Ralescu [3].

Definition 2.7 (see [3]). Let be a module over a ring and be a Complete Heyting algebra. An subset in is called an -submodule of , if for every and the following conditions are satisfied:(1),(2),(3).We denote the set of all -submodules of by . If , then is called a fuzzy submodule of . The set of all fuzzy submodules of are denoted by .

Definition 2.8 (see [9]). Let be such that . Then the quotient of with respect to , is a fuzzy submodule of , denoted by , and is defined as follows: where denotes the coset .

Definition 2.9 (see [9]). Let . Then is a fuzzy submodule of , and it is called the fuzzy submodule generated by the fuzzy subset . We denote this by , that is, Let . If for some , then is called a generating fuzzy subset of .

Remark 2.10. (a) If is a nonempty subset of , then , where is the submodule of generated by .
(b) If , then is a fuzzy submodule of generated by , and in this case,

#### 3. Preliminaries

This section contains some preliminary results that are needed in the sequel.

Lemma 3.1 (see [10]). Let be a module and suppose that and . Then(a) if and only if and ;(b)if , then ;(c)if is a direct summand of , then if and only if ;(d)if and for , then if and only if and .

Lemma 3.2 (see [9]). Let . Then .

Lemma 3.3 (see [9]). Let , for each , where . Then and .

Lemma 3.4 (see [5]). Let . Then the level subset , is a submodule of if and only if is a fuzzy submodule of .

Corollary 3.5. is a submodule of if and only if is a fuzzy submodule of .

In the next two sections we present our main results.

#### 4. Fuzzy Small Submodule

Definition 4.1. Let be a module over a ring and let . Then is said to be a Small -Submodule of , if for any satisfying implies . The notation indicates that is a small -submodule of .
If , then is called a fuzzy small submodule of and it is indicated by the notation . It is obvious that is always a fuzzy small submodule of .

Definition 4.2. Let . If , then we say (or is a fuzzy small cover of or .

Definition 4.3. Let and be any two modules over a ring . Then an ephimorphism is called a fuzzy small ephimorphism, if .

It is obvious that is a fuzzy small cover of if and only if the canonical projection is a fuzzy small ephimorphism.

Definition 4.4. A fuzzy ideal of with is called a fuzzy small ideal of if it is a fuzzy small submodule of .

Let and be any two fuzzy submodules of such that , then is called a fuzzy submodule of . And is called a fuzzy small submodule in , denoted by , if in the sense that for every submodule in satisfying implies (by we mean the restriction mapping of , on resp.).

Definition 4.5. Let be a module over a ring and suppose that with . Then we say lies above in or is a coessential extension of if is a fuzzy small cover of , that is, =.

Example 4.6. Consider under addition modulo 8. Then is a module over the ring . Let . Define as follows: where . Then is a fuzzy small submodule of .

Remark 4.7. Let . Clearly are the only proper submodules of and . Therefore . Also . Moreover, if we take then becomes the characteristic function of .

The above remark inspires us to state the following two theorems.

Theorem 4.8. Let be a module and . Then if and only if .

Proof. Let . We assume is not a fuzzy small submodule of . Thus there exists, , such that . Let . Then So, there exist with such that . Thus we have and , and so , . This implies that . Since is arbitrary, so this implies . But . So, we must have and this implies , a contradiction. Therefore, .
Conversely, we assume . If possible let be not a small submodule of . Thus there exists , , but . Thus and , . Let . Since , so there exist, , such that . Now, This implies and it contradicts the fact that . Hence .

Here we present an alternative proof of the following theorem.

Theorem 4.9 (see [8]). Let . Then if and only if .

Proof. Suppose, . Let and . We claim . Now, implies . Since , so we must have . there exists such that ,, either or , for all , , either or , for all , ,,,.Conversely, we assume . Let be such that . This implies that . Thus (since . This implies that there exists such that . Thus for every with implies either or otherwise . Therefore,
or for every and ,, for every and ,,,,.

Corollary 4.10. Let . Then if and only if .

Proof. Let . So, by above theorem we get . Conversely if . So, by above theorem . Hence .

Theorem 4.11 (see [8]). Let . Then , if and only if .

As a consequence, we obtain the following.

Theorem 4.12. Any finite sum of fuzzy small submodules of is also a fuzzy small submodule in .

Theorem 4.13. Let and be such that . If , then .

Proof. Let be such that . Let . Then
,=,=,= (since ). Therefore, . Since , so . This implies that and . Thus . Hence .

Corollary 4.14. Let and . If , then .

Proof. By definition, means . Therefore, from above theorem we get .

Theorem 4.15. Let . Then if and only if .

Proof. Suppose, . Let and . This implies . Since , so . This ensures that there exists in such that . Thus and hence .
Conversely, we assume . This implies . Therefore, (Lemma 3.1(b)). So, by Corollary 4.10 we have .

Definition 4.16. A fuzzy submodule in is called a fuzzy direct sum of two fuzzy submodules and if and .

Definition 4.17. Any is called a fuzzy direct summand of if there exists such that is a fuzzy direct sum of .

Theorem 4.18. Let be fuzzy submodules of M with and be a direct summand of . Then if and only if .

Proof. Suppose, . Since is a direct summand of , so there exists such that First we prove . Now, implies . Also, from we have . We claim . For this let . Then
, and for some , ,, for some , ,,, On the other hand if , then for some with , . This implies . Thus and so, we have and hence . Therefore, is a direct summand of . But, we know if and only if . Thus and is a direct summand of and so, by Lemma 3.1(c) we get . Hence, by Corollary 4.10 we have .
The proof of the converse part follows from Corollary 4.14.

Theorem 4.19. Let and . Also, let be such that and . Then and if and only if , that is, if and only if .

Proof. Suppose, and . Then and (Corollary 4.10). Therefore, we get (Lemma 3.1(d)). Now, implies and since , therefore we get . Hence (Corollary 4.10).
Conversely, we assume if and only if . Now, and imply . Again, since is a fuzzy direct summand of and , so by Theorem 4.18 we get . Similarly, it can be proved that .

Corollary 4.20. Let , and . Let , . Define for all , . Then if and only if and .

Theorem 4.21 (see [8]). Let , be any two modules over the same ring R and let be a module homomorphism. If , then .

Theorem 4.22. Let be such that . Then if and only if and , that is, if and only if and .

Proof. It is obvious that, . So, it is sufficient to show if and only if and .
Suppose, . Then since , so . Next, we will prove . Consider, the natural homomorphism, , defined by , where denotes the coset . Since , so by Theorem 4.21 we get, . Now, =,=,=,=,=,=,=,= for all .Thus on and . Therefore, .
Conversely, we assume and . To show , let be such that . Since and , so we have . This implies =. Again, since and , so we must have,,,, (see [9, Theorem ]), (since ), (since ). Therefore, .

Corollary 4.23. Let be such that . If , then is a coessential extension of in .

Theorem 4.24. Let be such that . Then is a coessential extension of in if and only if holds for all with .

Proof. Suppose, is a coessential extension of , that is, . If for all with , then So, (since ). Thus .
Conversely, we assume holds for all with . If there exists containing such that , then . This yields , (since ) and so . Hence is a coessential extension of .

As a consequence, we obtain the following.

Theorem 4.25. Let be such that . Then is a coessential extension of in if and only if is a coessential extension of in and is a coessential extension of in .

#### 5. Jacobson -Radical

Definition 5.1. Let . Then is called a maximal -submodule of if (i.e., is proper -submodule of ) and if any other proper -submodules of containing , then . Equivalently is a maximal element in the set of all nonconstant -submodules of under point wise partial ordering.

The intersection of all maximal -submodules of is known as Jacobson -radical of and is denoted by .

Theorem 5.2. Let . Then is a maximal -submodule of if and only if can be expressed as , where is a maximal submodule of and is a maximal element of -.

Proof. Proof is similar to the proof of Theorem of [9] and so it is omitted.

Lemma 5.3. Let M be a module over R and let M. Then if and only if is in the sum of all small -submodules of M.

Proof. Suppose, . Then by Theorem 4.9 we have, . But, from Remark 2.10(b) we have Therefore, as . This implies is in the sum of all small -submodules of .
Conversely, we assume is in the sum of all small -submodules of . Then (finite), where and so, , where . Now, for all . Therefore, (since ). Therefore, by Theorem 4.9 we have . This implies . But, , and so, we must have, . Since . Therefore, we have .

Definition 5.4 (see [9]). Let be a complete Heyting algebra. Then - is called a maximal element, if there does not exist - such that .

Theorem 5.5. For any module , (the Jacobson -radical of ), is the sum of all small -submodules of .

Proof. In view of Lemma 5.3, it is sufficient to show that if and only if ; or equivalently: is not a subset of if and only if is not a small -submodule of . We will proof the later one.
Suppose, is not a subset of . Then there exists a maximal -submodule of such that is not a subset of . This implies which is not a submodule of . Since is maximal, so . Therefore, is a maximal -submodule of which is properly contained in . This implies . Thus and . So, by definition we have, is not a small -submodule of .
Conversely, we assume is not a small -submodule of . Then there exists a with such that . Let be the collection of all such . Then because, . Now, for each , and is not a subset of . Moreover, any proper fuzzy submodule containing is also in . Also, forms a poset and the union of members of a chain in is again a member of . Therefore, by Zorn's lemma, has a maximal element, (say) . Since so, is not a subset of . Now, let be such that . If , then is also in . So, by maximality of , we have . This shows that is a maximal -submodule of . Since is not a subset of , so we must have is not a subset of .

Corollary 5.6. If is a small -submodule of , then it is the largest small -submodule of .

Corollary 5.7. Let be finitely generated module, then exists and is a small -submodule of provided - has a maximal element.

Remark 5.8. However, if we take , then - does not possess any maximal element, and so by Theorem 5.2, maximal -submodule of does not exist. Since the existence of the depends on the existence of maximal -submodule of , therefore, the assumption of the existence of a maximal element of - in corollary 5.5 is necessary.

Example 5.9. Let . Then is a Complete Heyting algebra together with the operations minimum(meet), maximum(join) and ≤(partial ordering), then 0.75 is a maximal element of -. Consider under addition modulo 8. Then is a module over the ring . Let . Define as follows: Then , which is a maximal submodule of . Also, , where 0.75 is a maximal element of -. So, by Theorem 5.2 we have as a maximal -submodule of . In fact, is the only maximal -submodule of and so . Since and hence by Theorem 4.9, we get . Thus . However, if we consider , then [0,1) does not have a maximal element and so by Theorem 5.2 there does not exist any maximal -submodule(maximal fuzzy submodule). So, does not exist when .

#### 6. Conclusion

In this paper some aspects and properties of fuzzy small submodules have been introduced which dualize the notion of fuzzy essential submodules. This concept has opened a new avenue toward the study of fuzzy Goldie dimension, for example using the notion of fuzzy small submodules one can define hollow submodules and discrete submodules. In our future study we may investigate various aspects of (i) spanning dimension of fuzzy submodules, (ii) corank of fuzzy submodules, (iii) fuzzy lifting modules with chain condition on fuzzy small submodules, and (iv) Noetherian and Artinian conditions on fuzzy Jacobson radical of a module.

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