#### Abstract

The aim of the paper is to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties. In order to provide these soft algebraic structures, the notion of -soft union rings which is a generalization of that of soft union rings is proposed. By introducing the notion of soft cosets, soft quotient rings based on -soft union ideals are established. Moreover, through discussing quotient soft subsets, an approach for constructing quotient soft union rings is made. Finally, isomorphism theorems of -soft union rings related to invariant soft sets are discussed.

#### 1. Introduction

Fuzzy set theory [1], intuitionistic set theory [2], and probability theory are useful approaches to describe uncertainty, but each of these theories has its inherent difficulties. To overcome these problems, Molodtsov [3] initiated the concept of soft sets that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. Maji et al. [4] gave the operations of soft sets and their properties; furthermore, they [5] introduced fuzzy soft sets which combine the strengths of both soft sets and fuzzy sets. As a generalization of the soft set theory, the fuzzy soft set theory makes description of the objective world more realistic, practical, and precise in some cases, making it very promising. Since its introduction, the concept of soft sets has gained considerable attention in many directions and has found applications in a wide variety of fields such as the theory of soft sets [6, 7] and soft decision making [8, 9].

Since the notion of soft groups was proposed by Aktaş and Çaǧman [10], then the soft set theory is used as a new tool to discuss algebraic structures. Acar et al. [11] initiated the concepts of soft rings similar to soft groups. Liu et al. further investigated isomorphism and fuzzy isomorphism theories of soft rings in [12, 13], respectively. Soft sets were also applied to other algebraic structures such as near-rings [14], -hyperrings [15], -modules [16, 17], and BCK/BCI-algebras [18]. The idea of quasicoincidence of a fuzzy point with a fuzzy set, which is mentioned in [19], has played a vital role in generating some different algebraic structures. By using the concepts of belongingness to (denoted by ) and quasicoincidence (denoted by ) of a fuzzy point with a fuzzy subgroup, Bhakat and Das [20] proposed the concept of -fuzzy subgroups. Inspired by the previous works, Zhan et al. [21] extended these results to BCI-algebras and obtained some important and useful generalizations of related algebraic structures. Moreover, they characterized filteristic soft BL-algebras [22] and filteristic soft MTL-algebras [23] based on -soft sets and -soft sets.

Çağman et al. [24] studied on soft int-groups, which are different from the definition of soft groups [10]. The new approach is based on the inclusion relation and intersection of sets. It brings the soft set theory, the set theory, and the group theory together. On the basis of soft int-groups, Sezgin et al. [25] introduced the concept of soft intersection near-rings (soft int near-rings) by using intersection operation of sets and gave the applications of soft int near-rings to the near-ring theory. By introducing soft intersection-union products and soft characteristic functions, Sezer [26] made a new approach to the classical ring theory via the soft set theory, with the concepts of soft union rings, ideals, and bi-ideals. Jun et al. applied intersectional soft sets to BCK/BCI-algebras [27, 28] and obtained many results.

In the present paper, in order to further investigate the application of soft sets in the ring theory, we introduce the notions of -soft union rings and -soft union ideals as generalizations of that of soft union rings and soft union ideals, respectively. Then, we discuss the properties of images and inverse images of -soft union ideals. Furthermore, we establish soft quotient rings based on -soft union ideals by introducing the notion of soft cosets. Moreover, through discussing quotient soft subsets, we give an approach for constructing quotient soft union rings. Finally, we discuss isomorphism theorems of -soft union rings related to invariant soft sets.

#### 2. Preliminaries

In this section, we would like to recall some basic notions related to soft sets and soft union rings. An algebraic system is called a ring if it satisfies the following conditions:(1) forms an abelian group,(2) forms a semigroup,(3) and , for all .

A subgroup of with and is called an ideal of .

Throughout the paper, , and denote rings, and , , and are the zero elements of , , and , respectively. is an initial universe and is a set of parameters under consideration with respect to . and are subsets of . The set of all subsets of is denoted by . Molodtsov [3] defined the concept of soft sets in the following way.

Definition 1 (see [3]). A soft set over is defined as such that if .

In other words, a soft set over is a parameterized family of subsets of the universe . For all , may be considered as the set of -approximate elements of the soft set . A soft set over can be presented by the set of ordered pairs: Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [3].

If is a soft set over , then the image of is defined by . The set of all soft sets over will be denoted by . Some of the operations of soft sets are listed as follows.

Definition 2 (see [4]). Let . If , for all , then is called a soft subset of and denoted by .
and are called soft equal, denoted by , if and only if and .

Definition 3 (see [24]). Let and let be a function from to . Then, the soft anti-image of under , denoted by , is a soft set over defined by for all . And the soft preimage of under , denoted by , is a soft set over defined by , for all .

Note that the concept of level sets in the fuzzy set theory, Çağman et al. [24] initiated the concept of lower inclusions of soft sets which serves as a bridge between soft sets and crisp sets.

Definition 4 (see [26]). Let be a soft set over and . Then, lower -inclusion of , denoted by , is defined as .

Inspired by the concept of soft int-groups [24], Sezer in [26] introduced the concept of soft union rings by the combination of the theories of soft sets and rings.

Definition 5 (see [26]). A soft set over is called a soft union ring of if (1),(2),(3),for all .

Now, we proceed on to recall the notion of soft union ideals of rings.

Definition 6 (see [26]). A soft set over is called a soft union left (resp., right) ideal of if (1),(2)  (resp., ),for all .
A soft set over is called a soft union ideal of if it is both a soft union left and a soft union right ideal of over .

#### 3. -Soft Union Rings and -Soft Ideals of Rings

In this section, we introduce the notion of -soft union rings (ideals) which is a generation of that of soft union rings (ideals) and investigate their basic properties. From now on, unless otherwise specified.

Definition 7. Let be a soft set over . is called a -soft union ring of if (1),(2),(3),for all .

Remark 8. Let be a soft union ring of over ; then is a -soft union ring of . Therefore, a soft union ring of is a -soft union ring, but the converse is not true in general.

We show this fact by the following example.

Example 9. Given a ring with the operations addition and multiplication of matrices, . We define a soft set over by Then, one can easily show that is a -soft union ring of . But is not a soft union ring of because

In order to give some characterizations of -soft union rings, we need the following lemmas.

Lemma 10. Let be a soft set over . If is a -soft union ring of , then (1),(2),for all .

Proof. Assume that is a -soft union ring of ; then for all , we get that  .
It is straightforward.

Lemma 11. Let be a soft set over . If is a -soft union ring of , then (1),(2),for all .

Proof. Assume that is a -soft union ring of ; then for all , we have that , by Lemma 10.
It is straightforward.

Combining Lemma 11 and Definition 7, we obtain the following characterization of -soft union rings.

Theorem 12. Let be a soft set over . Then, is a -soft union ring of if and only if satisfies the following conditions: (1),(2),for all .

Proof. Assume that is a -soft union ring of . By Definition 7 and Lemma 11, we have and for all .
Conversely, since , for all , by Lemma 11, we get Moreover, we have And it follows from hypothesis that ; therefore is a -soft union ring of .

Analogues to the notion of -soft union rings, we can extend the concept of soft union ideals as follows.

Definition 13. A soft set over is called a -soft union ideal of if (1),(2),for all .

Remark 14. Let be a soft union ideal of over ; then is a -soft union ideal of . Therefore, a soft union ideal of is a -soft union ideal; however it is important and interesting to note that the converse is not true in general.

Example 15. Let be a ring and a universal set. We construct a soft set over by , , and . One can show that is a -soft union ideal. But is not a soft union ideal of , since .

Proposition 16. If is a -soft union ideal of , then is an ideal of .

Proof. We need to show that (i) , (ii) , and (iii) , for all and . If , then . By Lemma 11, we obtain that , , and , for all and . Since is a -soft union ideal, then for all and , we have Similarly, we can prove that , for all and . Hence, . Therefore, (i) , (ii) , and (iii)   , for all and . And thus, is an ideal of .

We will now display the relationship between -soft union ideals and ideals. For this purpose, we require the following notion.

Definition 17. Let be a -soft union ideal of ; then is called the extended image set of , where .

Now, we characterize -soft union ideals by lower inclusions.

Proposition 18. Let be a soft set over and a totally ordered set by inclusion. Then, is a -soft union ideal of if and only if is an ideal of , whenever it is nonempty, for each where .

Proof. Assume that is a -soft union ideal of and is nonempty. It is sufficient to show that , and , for all , and . Let . It follows that and . Since is a -soft union ideal of and is a totally ordered set, then , , and . And thus, , , and ; hence . Therefore, is an ideal of .
Conversely, assume that is an ideal of whenever it is nonempty, for each where . Suppose that does not hold for some ; then there exist such that . Therefore, and . It follows that , but ; that is, , which is a contradiction. Hence, , for all . Similarly, we can prove that , for all . Thus, is a -soft union ideal of .

In the rest of this section, we will show that the soft anti-image and soft preimage of a -soft union ideal under a ring homomorphism are also -soft union ideals.

Theorem 19. Let be a soft set over and a ring epimorphism from to . If is a -soft union ideal of , then is a -soft union ideal of and .

Proof. Let and a -soft union ideal of . Since is a ring epimorphism from to , then and . And thus, there exist such that , . Therefore, we have Therefore, is a -soft union ideal of .
By Lemma 11, we have = .

Theorem 20. Let be a soft set over and a ring homomorphism from to . If is a -soft union ideal of , then is a -soft union ideal of .

Proof. Let . Then,
Moreover, we have
Hence, is a -soft union ideal of .

#### 4. Soft Quotient Rings

The main purpose of this section is to give an approach for constructing soft quotient rings based on -soft union ideals. Such approach involves the concept of soft cosets. In addition, some simple characterizations of soft cosets are presented.

Definition 21. Let be a -soft union ring of over and . Then, a soft coset of is defined by for all .

For the sake of simplicity and better understanding, we illustrate the above concept by the following example.

Example 22. Consider the -soft union ring in Example 9. If , then we define a soft set as Then, it is easy to show that is a soft coset of .

Proposition 23. Let be a -soft union ring over and . Then, if and only if .

Proof. Suppose that . Since is a -soft union ring, then = . By Lemma 11, we have . Thus, .
Conversely, assume that . We can prove that in a similar way.

Proposition 24. Let be a -soft union ring over and . Then, if and only if .

Proof. Suppose that ; then for all . Therefore, . Similarly, we can show that . Hence, .
Conversely, assume that . It follows that = .

Based on the above proposition, we give a property related to soft cosets as follows.

Proposition 25. Let be a -soft union ideal over and . If

Proof. Suppose that , . Then, and , by Proposition 24. Since is a -soft union ideal, then On the other hand, it follows from Lemma 11 that . Hence, , and so .
Moreover, According to Lemma 11, we get that . Therefore, ; that is, .

In view of Proposition 25, we have the following result.

Proposition 26. Let be a -soft union ideal over . Then, is a ring, where , , and , for all .

Proof. It is straightforward.

Definition 27. Let be a -soft union ideal over . Then, is called a soft quotient ring.

Theorem 28. Let be a -soft union ideal over . Then, .

Proof. Assume that such that , for all . It is easy to see that is a surjective homomorphism from to . Since = , therefore .

#### 5. Quotient Soft Union Rings

In this section, quotient soft union rings are constructed by introducing the notion of quotient soft subsets and some isomorphism theorems of -soft union rings related to invariant soft sets are discussed.

Definition 29. Let be a -soft union ideal over and a soft set. Then, a quotient soft subset of is defined by for all .

We will illustrate the above concept by the following example.

Example 30. Consider the -soft union ideal over in Example 15. We construct a soft set over by , , , and . We define a soft set such that , . Then, one can show that is a quotient soft subset of .

Proposition 31. Let be a -soft union ideal of over . Then, is a -soft union ideal of , where , for all .

Proof. It is easy to see that is a soft set. For all , we have
Moreover, since = , for all , then is a -soft union ideal of .

Proposition 32. Let be a -soft union ideal and a soft union ring. Then, is a soft union ring of .

Proof. For all , we have
Similarly, we can show that , for all . Thus, is a soft union ring of .

Definition 33. Let be a -soft union ideal and a soft union ring. Then, is called a quotient soft union ring of .

In order to investigate isomorphism theorems of -soft union rings, we need a concept of invariant soft sets which will play an important role in the sequel.

Definition 34. Let be a ring homomorphism. A soft set over is called an invariant soft set with respect to if implies , for all .

Proposition 35. Let be a ring homomorphism and a soft set of over . Then, is an invariant soft set with respect to .

Proof. Let such that . Then, = . Hence, is an invariant soft set with respect to .

Next, we establish isomorphism theorems of -soft union rings.

Theorem 36 (first isomorphism theorem). Let be an epimorphism and let -soft union ideal be a invariant soft set with respect to . Then, .

Proof. Let be a mapping such that , for all . Obviously, is an epimorphism. Since is an invariant soft set with respect to , then . Therefore, . By Theorem 28, we have . Hence, .

Proposition 37. Let be an epimorphism and -soft union ideal and a invariant soft set with respect to . Then, .

Proof. It follows from Theorem 28 and Proposition 35 that is a -soft union ideal of and is an invariant soft set with respect to . Since is an epimorphism, then . By Theorem 36, we get that .

Theorem 38. Let be an epimorphism, a soft union ring of , and a -soft union ideal of . If is an invariant soft set with respect to , then .

Proof. Let be a mapping such that , for all . By Theorem 36, we have that is a ring isomorphism. Let . Since is a subjective homomorphism, then there exists such that . Considering that is an invariant soft set with respect to , we obtain Hence, , and so .

Proposition 39. Let be an epimorphism. If is a soft union ring of and is a -soft union ideal of , then .

Proof. Since is a subjective homomorphism, then and . By Proposition 35, we get that is an invariant soft set with respect to . It follows from Theorem 38 that .

Lemma 40. Let and be -soft union ideals of such that and . Then, if and only if , for all .

Proof. Assume that , for all . Then, we have that , and so
If , then . Therefore, It follows that = . On the other hand, we have , by Lemma 11. Hence, . And thus, .
Conversely, suppose that , for all . It follows that . Therefore, that is, . It follows from Proposition 24 that .

Theorem 41 (third isomorphism theorem). Let and be -soft union ideals of such that and . If is a soft union ring, then .

Proof. Let be a mapping such that , for all . According to Lemma 40, we obtain that is an injective mapping. Obviously, is subjective. For all , we have Thus, is a ring isomorphism.
Moreover, since that is, therefore , and so .

#### 6. Conclusions

We introduced the notions of -soft union rings and -soft union ideals which are generations of that of soft union rings and soft union ideals in the paper. Then, we established soft quotient rings based on -soft union ideals by using soft cosets and gave an approach for constructing quotient soft union rings. Finally, we considered isomorphism theorems of -soft union rings related to invariant soft sets. Motivated by the notion of soft intersection near-rings in [25], we will investigate the properties of -soft near-rings. In addition, it is interesting to apply the theory of soft union (intersection) sets to other algebraic structures such as modules.

#### Acknowledgments

The authors would like to thank the editors and referees for their constructive comments in improving this paper significantly. The works described in this paper are partially supported by grants from Graduate Independent Innovation Foundation of Northwest University (YZZ12061) and Graduate Higher Achievement Foundation of Northwest University (YC13055).