Applications of Soft Union Sets in the Ring Theory

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 (𝜆,𝜇)-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 (𝜆,𝜇)-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 (𝜆,𝜇)-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], Γ-hyperrings [15], Γ-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 𝑞) of a fuzzy point with a fuzzy subgroup, Bhakat and Das [20] proposed the concept of (𝛼,𝛽)-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 𝑞-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 (𝜆,𝜇)-soft union rings and (𝜆,𝜇)-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 (𝜆,𝜇)-soft union ideals. Furthermore, we establish soft quotient rings based on (𝜆,𝜇)-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 (𝜆,𝜇)-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 (𝑅,+,⋅) is called a ring if it satisfies the following conditions:(1)(𝑅,+) forms an abelian group,(2)(𝑅,⋅) forms a semigroup,(3)𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐 and (𝑏+𝑐)𝑎=𝑏𝑎+𝑐𝑎, for all 𝑎,𝑏,𝑐∈𝑅.

A subgroup 𝑆 of (𝑅,+) with 𝑆𝑅⊆𝑆 and 𝑅𝑆⊆𝑆 is called an ideal of 𝑅.

Throughout the paper, 𝑅,𝑅₁, and 𝑅₂ denote rings, and 0, 0₁, and 0₂ are the zero elements of 𝑅, 𝑅₁, and 𝑅₂, respectively. 𝑈 is an initial universe and 𝐸 is a set of parameters under consideration with respect to 𝑈. 𝐴 and 𝐵 are subsets of 𝐸. The set of all subsets of 𝑈 is denoted by 𝒫(𝑈). Molodtsov [3] defined the concept of soft sets in the following way.

Definition 1 (see [3]). A soft set 𝑓_𝐴 over 𝑈 is defined as 𝑓_𝐴∶𝐸→𝒫(𝑈) such that 𝑓_𝐴(𝑥)=∅ if 𝑥∉𝐴.

In other words, a soft set over 𝑈 is a parameterized family of subsets of the universe 𝑈. For all 𝜀∈𝐴, 𝑓_𝐴(𝜀) may be considered as the set of 𝜀-approximate elements of the soft set 𝑓_𝐴. A soft set 𝑓_𝐴 over 𝑈 can be presented by the set of ordered pairs: 𝑓_𝐴=𝑥,𝑓_𝐴(𝑥)∣𝑥∈𝐸,𝑓_𝐴(𝑥)∈𝒫(𝑈). (1) Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [3].

If 𝑓_𝐴 is a soft set over 𝑈, then the image of 𝑓_𝐴 is defined by Im(𝑓_𝐴)={𝑓_𝐴(𝑎)∣𝑎∈𝐴}. The set of all soft sets over 𝑈 will be denoted by 𝑆(𝑈). Some of the operations of soft sets are listed as follows.

Definition 2 (see [4]). Let 𝑓_𝐴,𝑓_𝐵∈𝑆(𝑈). If 𝑓_𝐴(𝑥)⊆𝑓_𝐵(𝑥), for all 𝑥∈𝐸, then 𝑓_𝐴 is called a soft subset of 𝑓_𝐵 and denoted by 𝑓_𝐴⊆𝑓_𝐵.
𝑓_𝐴 and 𝑓_𝐵 are called soft equal, denoted by 𝑓_𝐴=𝑓_𝐵, if and only if 𝑓_𝐴⊆𝑓_𝐵 and 𝑓_𝐵⊆𝑓_𝐴.

Definition 3 (see [24]). Let 𝑓_𝐴,𝑓_𝐵∈𝑆(𝑈) and let Ψ be a function from 𝐴 to 𝐵. Then, the soft anti-image of 𝑓_𝐴 under Ψ, denoted by Ψ(𝑓_𝐴), is a soft set over 𝑈 defined by Ψ𝑓_𝐴(𝑏)=∩𝑓_𝐴(𝑎)∣𝑎∈𝐴,Ψ(𝑎)=𝑏,ifΨ⁻¹(𝑏)≠∅,∅,otherwise, (2) for all 𝑏∈𝐵. And the soft preimage of 𝑓_𝐵 under Ψ, denoted by Ψ⁻¹(𝑓_𝐵), is a soft set over 𝑈 defined by Ψ⁻¹(𝑓_𝐵)(𝑎)=𝑓_𝐵(Ψ(𝑎)), for all 𝑎∈𝐴.

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 𝑓_𝐴 be a soft set over 𝑈 and 𝛼⊆𝑈. Then, lower 𝛼-inclusion of 𝑓_𝐴, denoted by 𝐿(𝑓_𝐴;𝛼), is defined as 𝐿(𝑓_𝐴;𝛼)={𝑥∈𝐴∣𝑓_𝐴(𝑥)⊆𝛼}.

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 𝑓_𝑅 over 𝑈 is called a soft union ring of 𝑅 if (1)𝑓_𝑅(𝑥+𝑦)⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦),(2)𝑓_𝑅(𝑥)⊆𝑓_𝑅(−𝑥),(3)𝑓_𝑅(𝑥𝑦)⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦),for all 𝑥,𝑦∈𝑅.

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

Definition 6 (see [26]). A soft set 𝑓_𝑅 over 𝑈 is called a soft union left (resp., right) ideal of 𝑅 if (1)𝑓_𝑅(𝑥−𝑦)⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦),(2)𝑓_𝑅(𝑥𝑦)⊆𝑓_𝑅(𝑦)  (resp., 𝑓_𝑅(𝑥𝑦)⊆𝑓_𝑅(𝑥)),for all 𝑥,𝑦∈𝑅.
A soft set 𝑓_𝑅 over 𝑈 is called a soft union ideal of 𝑅 if it is both a soft union left and a soft union right ideal of 𝑅 over 𝑈.

3. (𝜆,𝜇)-Soft Union Rings and (𝜆,𝜇)-Soft Ideals of Rings

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

Definition 7. Let 𝑓_𝑅 be a soft set over 𝑈. 𝑓_𝑅 is called a (𝜆,𝜇)-soft union ring of 𝑅 if (1)𝑓_𝑅(𝑥+𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇,(2)𝑓_𝑅(𝑥)∩𝜆⊆𝑓_𝑅(−𝑥)∪𝜇,(3)𝑓_𝑅(𝑥𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇,for all 𝑥,𝑦∈𝑅.

Remark 8. Let 𝑓_𝑅 be a soft union ring of 𝑅 over 𝑈; then 𝑓_𝑅 is a (𝑈,∅)-soft union ring of 𝑅. Therefore, a soft union ring of 𝑅 is a (𝜆,𝜇)-soft union ring, but the converse is not true in general.

We show this fact by the following example.

Example 9. Given a ring 𝑅={^𝑎𝑏₀₀∣𝑎,𝑏∈ℤ₂} with the operations addition and multiplication of matrices, 𝑈=ℤ₅. We define a soft set 𝑓_𝑅 over 𝑈 by 𝑓_𝑅={}00000,2,𝑓_𝑅={},𝑓01000,1,2,4_𝑅={}10000,1,2,𝑓_𝑅={}.11000,1,4 (3) Then, one can easily show that 𝑓_𝑅 is a ({0,1,2,4},{0,2})-soft union ring of 𝑅. But 𝑓_𝑅 is not a soft union ring of 𝑅 because 𝑓_𝑅[phone]=𝑓_𝑅={}00000,2⊆^/𝑓_𝑅1100∪𝑓_𝑅={}.11000,1,4 (4)

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

Lemma 10. Let 𝑓_𝑅 be a soft set over 𝑈. If 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅, then (1)(𝑓_𝑅(−𝑥)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥)∩𝜆)∪𝜇,(2)𝑓_𝑅(−𝑥)∩𝜆⊆𝑓_𝑅(𝑥)∪𝜇,for all 𝑥∈𝑅.

Proof. (1) Assume that 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅; then for all 𝑥∈𝑅, we get that  (𝑓_𝑅(−𝑥)∩𝜆)∪𝜇=(𝑓_𝑅(−𝑥)∩𝜆∩𝜆)∪𝜇⊆((𝑓_𝑅(−(−𝑥))∪𝜇)∩𝜆)∪𝜇=(𝑓_𝑅(𝑥)∩𝜆)∪𝜇.
(2) It is straightforward.

Lemma 11. Let 𝑓_𝑅 be a soft set over 𝑈. If 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅, then (1)(𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥)∩𝜆)∪𝜇,(2)𝑓_𝑅(0)∩𝜆⊆𝑓_𝑅(𝑥)∪𝜇,for all 𝑥∈𝑅.

Proof. (1) Assume that 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅; then for all 𝑥∈𝑅, we have that (𝑓_𝑅(0)∩𝜆)∪𝜇=(𝑓_𝑅(𝑥−𝑥)∩𝜆∩𝜆)∪𝜇⊆((𝑓_𝑅(𝑥)∪𝑓_𝑅(−𝑥)∪𝜇)∩𝜆)∪𝜇=((𝑓_𝑅(𝑥)∩𝜆)∪𝜇)∪(𝑓_𝑅(−𝑥)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥)∩𝜆)∪𝜇, by Lemma 10.
(2) It is straightforward.

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

Theorem 12. Let 𝑓_𝑅 be a soft set over 𝑈. Then, 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅 if and only if 𝑓_𝑅 satisfies the following conditions: (1)𝑓_𝑅(𝑥−𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇,(2)𝑓_𝑅(𝑥𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇,for all 𝑥,𝑦∈𝑅.

Proof. Assume that 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅. By Definition 7 and Lemma 11, we have 𝑓_𝑅(𝑥𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇 and 𝑓_𝑅𝑥−𝑦∩𝜆=𝑓_𝑅⊆𝑓𝑥−𝑦∩𝜆∩𝜆_𝑅(𝑥)∪𝑓_𝑅=𝑓−𝑦∪𝜇∩𝜆_𝑅(𝑥)∩𝜆∪𝑓_𝑅−𝑦∩𝜆∪𝜇⊆𝑓_𝑅(𝑥)∪𝑓_𝑅𝑦∪𝜇, (5) for all 𝑥,𝑦∈𝑅.
Conversely, since 𝑓_𝑅(𝑥−𝑦)∩𝜆⊆(𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦))∪𝜇, for all 𝑥,𝑦∈𝑅, by Lemma 11, we get 𝑓_𝑅(𝑥)∩𝜆=𝑓_𝑅((⊆𝑓0−−𝑥))∩𝜆∩𝜆_𝑅(0)∪𝑓_𝑅()=𝑓−𝑥∪𝜇∩𝜆_𝑅(0)∩𝜆∪𝑓_𝑅()⊆𝑓−𝑥∪𝜇∩𝜆_𝑅(−𝑥)∪𝜇∪𝑓_𝑅(−𝑥)∪𝜇=𝑓_𝑅(−𝑥)∪𝜇. (6) Moreover, we have 𝑓_𝑅𝑥+𝑦∩𝜆=𝑓_𝑅⊆𝑓𝑥−−𝑦∩𝜆∩𝜆_𝑅(𝑥)∪𝑓_𝑅=𝑓−𝑦∪𝜇∩𝜆_𝑅(𝑥)∪𝜇∩𝜆∪𝑓_𝑅−𝑦∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅𝑦∪𝜇. (7) And it follows from hypothesis that 𝑓_𝑅(𝑥𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇; therefore 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring of 𝑅.

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

Definition 13. A soft set 𝑓_𝑅 over 𝑈 is called a (𝜆,𝜇)-soft union ideal of 𝑅 if (1)𝑓_𝑅(𝑥−𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇,(2)𝑓_𝑅(𝑥𝑦)∩𝜆⊆(𝑓_𝑅(𝑥)∩𝑓_𝑅(𝑦))∪𝜇,for all 𝑥,𝑦∈𝑅.

Remark 14. Let 𝑓_𝑅 be a soft union ideal of 𝑅 over 𝑈; then 𝑓_𝑅 is a (𝑈,∅)-soft union ideal of 𝑅. Therefore, a soft union ideal of 𝑅 is a (𝜆,𝜇)-soft union ideal; however it is important and interesting to note that the converse is not true in general.

Example 15. Let 𝑅=ℤ₄ be a ring and 𝑈=ℤ⁺ a universal set. We construct a soft set 𝑓_𝑅 over 𝑈 by 𝑓_𝑅(0)={1}, 𝑓_𝑅(1)=𝑓_𝑅(3)={2,3,4}, and 𝑓_𝑅(2)={3}. One can show that 𝑓_𝑅 is a ({1,2,4},{1,2})-soft union ideal. But 𝑓_𝑅 is not a soft union ideal of 𝑅, since 𝑓_𝑅(2−2)=𝑓_𝑅(0)={1}⊆^/𝑓_𝑅(2)∪𝑓_𝑅(2)={3}.

Proposition 16. If 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅, then 𝑓^∗_𝑅≜{𝑥∈𝑅∣(𝑓_𝑅(𝑥)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇} is an ideal of 𝑅.

Proof. We need to show that (i) 𝑥−𝑦∈𝑓^∗_𝑅, (ii) 𝑟𝑥∈𝑓^∗_𝑅, and (iii) 𝑥𝑟∈𝑓^∗_𝑅, for all 𝑥,𝑦∈𝑓^∗_𝑅 and 𝑟∈𝑅. If 𝑥,𝑦∈𝑓^∗_𝑅, then (𝑓_𝑅(𝑥)∩𝜆)∪𝜇=(𝑓_𝑅(𝑦)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇. By Lemma 11, we obtain that (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥−𝑦)∩𝜆)∪𝜇, (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑟𝑥)∩𝜆)∪𝜇, and (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥𝑟)∩𝜆)∪𝜇, for all 𝑥,𝑦∈𝑓^∗_𝑅 and 𝑟∈𝑅. Since 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal, then for all 𝑥,𝑦∈𝑓^∗_𝑅 and 𝑟∈𝑅, we have 𝑓_𝑅=𝑓𝑥−𝑦∩𝜆∪𝜇_𝑅⊆𝑓𝑥−𝑦∩𝜆∩𝜆∪𝜇_𝑅(𝑥)∪𝑓_𝑅𝑦=𝑓∪𝜇∩𝜆∪𝜇_𝑅(𝑥)∪𝑓∩𝜆∪𝜇_𝑅𝑦=𝑓∩𝜆∪𝜇_𝑅(0)𝑓∩𝜆∪𝜇,_𝑅(𝑟𝑥)∩𝜆∪𝜇=𝑓_𝑅()⊆𝑓𝑟𝑥∩𝜆∩𝜆∪𝜇_𝑅(𝑟)∩𝑓_𝑅(𝑥)⊆𝑓∪𝜇∩𝜆∪𝜇_𝑅(𝑥)∩𝜆∪𝜇=𝑓_𝑅(0)∩𝜆∪𝜇. (8) Similarly, we can prove that (𝑓_𝑅(𝑥𝑟)∩𝜆)∪𝜇⊆(𝑓_𝑅(0)∩𝜆)∪𝜇, for all 𝑥∈𝑓^∗_𝑅 and 𝑟∈𝑅. Hence, (𝑓_𝑅(𝑥−𝑦)∩𝜆)∪𝜇=(𝑓_𝑅(𝑟𝑥)∩𝜆)∪𝜇=(𝑓_𝑅(𝑥𝑟)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇. Therefore, (i) 𝑥−𝑦∈𝑓^∗_𝑅, (ii) 𝑟𝑥∈𝑓^∗_𝑅, and (iii)   𝑥𝑟∈𝑓^∗_𝑅, for all 𝑥,𝑦∈𝑓^∗_𝑅 and 𝑟∈𝑅. And thus, 𝑓^∗_𝑅 is an ideal of 𝑅.

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

Definition 17. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal of 𝑅; then 𝐸Im(𝑓_𝑅) is called the extended image set of 𝑓_𝑅, where 𝐸Im(𝑓_𝑅)=Im(𝑓_𝑅)∪{𝜆,𝜇}.

Now, we characterize (𝜆,𝜇)-soft union ideals by lower inclusions.

Proposition 18. Let 𝑓_𝑅 be a soft set over 𝑈 and 𝐸Im(𝑓_𝑅) a totally ordered set by inclusion. Then, 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅 if and only if 𝐿(𝑓_𝑅;𝛼) is an ideal of 𝑅, whenever it is nonempty, for each 𝛼⊆𝑈 where 𝜇⊆𝛼⊂𝜆.

Proof. Assume that 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅 and 𝐿(𝑓_𝑅;𝛼) is nonempty. It is sufficient to show that 𝑥−𝑦∈𝐿(𝑓_𝑅;𝛼), 𝑟𝑥∈𝐿(𝑓_𝑅;𝛼) and 𝑥𝑟∈𝐿(𝑓_𝑅;𝛼), for all 𝑥,𝑦∈𝐿(𝑓_𝑅;𝛼), and 𝑟∈𝑅. Let 𝑥,𝑦∈𝐿(𝑓_𝑅;𝛼). It follows that 𝑓_𝑅(𝑥)⊆𝛼 and 𝑓_𝑅(𝑦)⊆𝛼. Since 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅 and 𝐸Im(𝑓_𝑅) is a totally ordered set, then 𝑓_𝑅(𝑥−𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇⊆𝛼∪𝛼∪𝛼=𝛼⊂𝜆, 𝑓_𝑅(𝑟𝑥)∩𝜆⊆(𝑓_𝑅(𝑟)∩𝑓_𝑅(𝑥))∪𝜇⊆𝛼∪𝛼=𝛼⊂𝜆, and 𝑓_𝑅(𝑥𝑟)∩𝜆⊆(𝑓_𝑅(𝑥)∩𝑓_𝑅(𝑟))∪𝜇⊆𝛼∪𝛼=𝛼⊂𝜆. And thus, 𝑓_𝑅(𝑥−𝑦)⊆𝛼, 𝑓_𝑅(𝑟𝑥)⊆𝛼, and 𝑓_𝑅(𝑥𝑟)⊆𝛼; hence 𝑥−𝑦,𝑟𝑥,𝑥𝑟∈𝐿(𝑓_𝑅;𝛼). Therefore, 𝐿(𝑓_𝑅;𝛼) is an ideal of 𝑅.
Conversely, assume that 𝐿(𝑓_𝑅;𝛼) is an ideal of 𝑅 whenever it is nonempty, for each 𝛼⊆𝑈 where 𝜇⊆𝛼⊂𝜆. Suppose that 𝑓_𝑅(𝑥−𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇 does not hold for some 𝑥,𝑦∈𝑅; then there exist 𝑥₀,𝑦₀∈𝑅 such that 𝑓_𝑅(𝑥₀−𝑦₀)∩𝜆⊃𝛼=(𝑓_𝑅(𝑥₀)∪𝑓_𝑅(𝑦₀)∪𝜇). Therefore, 𝑓_𝑅(𝑥₀)∪𝑓_𝑅(𝑦₀)⊆𝛼 and 𝜇⊆𝛼⊂𝜆. It follows that 𝑥₀,𝑦₀∈𝐿(𝑓_𝑅;𝛼), but 𝑓_𝑅(𝑥₀−𝑦₀)⊃𝛼; that is, 𝑥₀−𝑦₀∉𝐿(𝑓_𝑅;𝛼), which is a contradiction. Hence, 𝑓_𝑅(𝑥−𝑦)∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅(𝑦)∪𝜇, for all 𝑥,𝑦∈𝑅. Similarly, we can prove that 𝑓_𝑅(𝑥𝑦)∩𝜆⊆(𝑓_𝑅(𝑥)∩𝑓_𝑅(𝑦))∪𝜇, for all 𝑥,𝑦∈𝑅. Thus, 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅.

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

Theorem 19. Let 𝑓_𝑅1 be a soft set over 𝑈 and Ψ a ring epimorphism from 𝑅₁ to 𝑅₂. If 𝑓_𝑅1 is a (𝜆,𝜇)-soft union ideal of 𝑅₁, then Ψ(𝑓_𝑅1) is a (𝜆,𝜇)-soft union ideal of 𝑅₂ and (Ψ(𝑓_𝑅1)(0₂)∩𝜆)∪𝜇=(𝑓_𝑅1(0₁)∩𝜆)∪𝜇.

Proof. Let 𝑦₁,𝑦₂∈𝑅₂ and 𝑓_𝑅1 a (𝜆,𝜇)-soft union ideal of 𝑅₁. Since Ψ is a ring epimorphism from 𝑅₁ to 𝑅₂, then Ψ⁻¹(𝑦₁)≠∅ and Ψ⁻¹(𝑦₂)≠∅. And thus, there exist 𝑥₁,𝑥₂∈𝑅₁ such that Ψ(𝑥₁)=𝑦₁, Ψ(𝑥₂)=𝑦₂. Therefore, we have Ψ𝑓_𝑅1𝑦₁−𝑦₂𝑓∩𝜆=∩_𝑅1𝑥₁−𝑥₂∣Ψ𝑥₁−𝑥₂=𝑦₁−𝑦₂𝑓∩𝜆=∩_𝑅1𝑥₁−𝑥₂∩𝜆∣Ψ𝑥₁−𝑥₂=𝑦₁−𝑦₂⊆∩𝑓_𝑅1𝑥₁∪𝑓_𝑅1𝑥₂∪𝜇∣Ψ𝑥₁=𝑦₁,Ψ𝑥₂=𝑦₂=∩𝑓_𝑅1𝑥₁∣Ψ𝑥₁=𝑦₁∪∩𝑓_𝑅1𝑥₂∣Ψ𝑥₂=𝑦₂𝑓∪𝜇=Ψ_𝑅1𝑦₁∪Ψ𝑓_𝑅1𝑦₂Ψ𝑓∪𝜇,_𝑅1𝑦₁𝑦₂𝑓∩𝜆=∩_𝑅1𝑥₁𝑥₂∣Ψ𝑥₁𝑥₂=𝑦₁𝑦₂𝑓∩𝜆=∩_𝑅1𝑥₁𝑥₂∩𝜆∣Ψ𝑥₁𝑥₂=𝑦₁𝑦₂⊆∩𝑓_𝑅1𝑥₁∩𝑓_𝑅1𝑥₂∪𝜇∣Ψ𝑥₁=𝑦₁,Ψ𝑥₂=𝑦₂=∩𝑓_𝑅1𝑥₁∣Ψ𝑥₁=𝑦₁∩∩𝑓_𝑅1𝑥₂∣Ψ𝑥₂=𝑦₂=Ψ𝑓∪𝜇_𝑅1𝑦₁∩Ψ𝑓_𝑅1𝑦₁∪𝜇. (9) Therefore, Ψ(𝑓_𝑅1) is a (𝜆,𝜇)-soft union ideal of 𝑅₂.
By Lemma 11, we have (Ψ(𝑓_𝑅1)(0₂)∩𝜆)∪𝜇=(∩{𝑓_𝑅1(𝑥)∣𝑥∈𝑅₁,Ψ(𝑥)=0₂}∩𝜆)∪𝜇 = ∩{(𝑓_𝑅1(𝑥)∩𝜆)∪𝜇∣𝑥∈𝑅₁,Ψ(𝑥)=0₂}=(𝑓_𝑅1(0₁)∩𝜆)∪𝜇.

Theorem 20. Let 𝑓_𝑅2 be a soft set over 𝑈 and Ψ a ring homomorphism from 𝑅₁ to 𝑅₂. If 𝑓_𝑅2 is a (𝜆,𝜇)-soft union ideal of 𝑅₂, then Ψ⁻¹(𝑓_𝑅2) is a (𝜆,𝜇)-soft union ideal of 𝑅₁.

Proof. Let 𝑥₁,𝑥₂∈𝑅₁. Then, Ψ⁻¹𝑓_𝑅2𝑥₁−𝑥₂∩𝜆=𝑓_𝑅2Ψ𝑥₁−𝑥₂∩𝜆=𝑓_𝑅2Ψ𝑥₁−Ψ𝑥₂∩𝜆⊆𝑓_𝑅2Ψ𝑥₁∪𝑓_𝑅2Ψ𝑥₂∪𝜇=Ψ⁻¹𝑓_𝑅2𝑥₁∪Ψ⁻¹𝑓_𝑅2𝑥₁∪𝜇. (10)
Moreover, we have Ψ⁻¹𝑓_𝑅2𝑥₁𝑥₂∩𝜆=𝑓_𝑅2Ψ𝑥₁𝑥₂∩𝜆=𝑓_𝑅2Ψ𝑥₁⋅Ψ𝑥₂⊆𝑓∩𝜆_𝑅2Ψ𝑥₁∩𝑓_𝑅2Ψ𝑥₁=Ψ∪𝜇₋₁𝑓_𝑅2𝑥₁∩Ψ⁻¹𝑓_𝑅2𝑥₁∪𝜇. (11)
Hence, Ψ⁻¹(𝑓_𝑅2) is a (𝜆,𝜇)-soft union ideal of 𝑅₁.

4. Soft Quotient Rings

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

Definition 21. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ring of 𝑅 over 𝑈 and 𝑟∈𝑅. Then, a soft coset 𝑟⊕𝑓_𝑅 of 𝑓_𝑅 is defined by 𝑟⊕𝑓_𝑅(𝑥)=𝑓_𝑅(𝑥−𝑟)∩𝜆∪𝜇, (12) for all 𝑥∈𝑅.

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

Example 22. Consider the ({0,1,2,4},{0,2})-soft union ring 𝑓_𝑅 in Example 9. If 𝑟=⁰¹₀₀, then we define a soft set 𝑟⊕𝑓_𝑅 as 𝑟⊕𝑓_𝑅={},00000,1,2,4𝑟⊕𝑓_𝑅={},01000,2𝑟⊕𝑓_𝑅={},10000,1,2,4𝑟⊕𝑓_𝑅={}.11000,1,2 (13) Then, it is easy to show that 𝑟⊕𝑓_𝑅 is a soft coset of 𝑓_𝑅.

Proposition 23. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ring over 𝑈 and 𝑎,𝑏∈𝑅. Then, (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇 if and only if (𝑓_𝑅(𝑏−𝑎)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇.

Proof. Suppose that (𝑓_𝑅(𝑏−𝑎)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇. Since 𝑓_𝑅 is a (𝜆,𝜇)-soft union ring, then (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(𝑎−𝑏)∩𝜆∩𝜆)∪𝜇⊆((𝑓_𝑅(𝑏−𝑎)∪𝜇)∩𝜆)∪𝜇=(𝑓_𝑅(𝑏−𝑎)∩𝜆)∪𝜇 = (𝑓_𝑅(0)∩𝜆)∪𝜇. By Lemma 11, we have (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇. Thus, (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇.
Conversely, assume that (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇. We can prove that (𝑓_𝑅(𝑏−𝑎)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇 in a similar way.

Proposition 24. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ring over 𝑈 and 𝑎,𝑏∈𝑅. Then, 𝑎⊕𝑓_𝑅=𝑏⊕𝑓_𝑅 if and only if (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇.

Proof. Suppose that (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇; then 𝑏⊕𝑓_𝑅(𝑥)=𝑓_𝑅()=𝑓𝑥−𝑏∩𝜆∪𝜇_𝑅()=𝑓𝑥−𝑎+𝑎−𝑏∩𝜆∪𝜇_𝑅()⊆𝑓𝑥−𝑎+𝑎−𝑏∩𝜆∩𝜆∪𝜇_𝑅(𝑥−𝑎)∪𝑓_𝑅()=𝑓𝑎−𝑏∪𝜇∩𝜆∪𝜇_𝑅(𝑥−𝑎)∩𝜆∪𝜇∪𝑓_𝑅()=𝑓𝑎−𝑏∩𝜆∪𝜇_𝑅(𝑥−𝑎)∩𝜆∪𝜇∪𝑓_𝑅(0)⊆𝑓∩𝜆∪𝜇_𝑅(𝑥−𝑎)∩𝜆∪𝜇=𝑎⊕𝑓_𝑅(𝑥), (14) for all 𝑥∈𝑅. Therefore, 𝑏⊕𝑓_𝑅⊆𝑎⊕𝑓_𝑅. Similarly, we can show that 𝑎⊕𝑓_𝑅⊆𝑏⊕𝑓_𝑅. Hence, 𝑎⊕𝑓_𝑅=𝑏⊕𝑓_𝑅.
Conversely, assume that 𝑎⊕𝑓_𝑅=𝑏⊕𝑓_𝑅. It follows that (𝑓_𝑅(𝑎−𝑏)∩𝜆)∪𝜇=(𝑏⊕𝑓_𝑅)(𝑎) = (𝑎⊕𝑓_𝑅)(𝑎)=(𝑓_𝑅(0)∩𝜆)∪𝜇.

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

Proposition 25. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal over 𝑈 and 𝑎,𝑏,𝑥,𝑦∈𝑅. If 𝑥⊕𝑓_𝑅=𝑎⊕𝑓_𝑅,𝑦⊕𝑓_𝑅=𝑏⊕𝑓_𝑅,𝑡ℎ𝑒𝑛𝑥+𝑦⊕𝑓_𝑅=(𝑎+𝑏)⊕𝑓_𝑅,𝑥𝑦⊕𝑓_𝑅=𝑎𝑏⊕𝑓_𝑅. (15)

Proof. Suppose that 𝑥⊕𝑓_𝑅=𝑎⊕𝑓_𝑅, 𝑦⊕𝑓_𝑅=𝑏⊕𝑓_𝑅. Then, (𝑓_𝑅(𝑥−𝑎)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇 and (𝑓_𝑅(𝑦−𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇, by Proposition 24. Since 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal, then 𝑓_𝑅−()=𝑓𝑥+𝑦𝑎+𝑏∩𝜆∪𝜇_𝑅()+=𝑓𝑥−𝑎𝑦−𝑏∩𝜆∪𝜇_𝑅)+⊆𝑓(𝑥−𝑎𝑦−𝑏∩𝜆∩𝜆∪𝜇_𝑅(𝑥−𝑎)∪𝑓_𝑅=𝑓𝑦−𝑏∪𝜇∩𝜆∪𝜇_𝑅(𝑥−𝑎)∩𝜆∪𝑓_𝑅=𝑓𝑦−𝑏∩𝜆∪𝜇_𝑅(0)∩𝜆∪𝜇. (16) On the other hand, it follows from Lemma 11 that (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅((𝑥+𝑦)−(𝑎+𝑏))∩𝜆)∪𝜇. Hence, (𝑓_𝑅((𝑥+𝑦)−(𝑎+𝑏))∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇, and so (𝑥+𝑦)⊕𝑓_𝑅=(𝑎+𝑏)⊕𝑓_𝑅.
Moreover, 𝑓_𝑅=𝑓𝑥𝑦−𝑎𝑏∩𝜆∪𝜇_𝑅()=𝑓𝑥−𝑎𝑦+𝑎𝑦−𝑏∩𝜆∪𝜇_𝑅)⊆𝑓(𝑥−𝑎𝑦+𝑎𝑦−𝑏∩𝜆∩𝜆∪𝜇_𝑅(𝑥−𝑎)𝑦∪𝑓_𝑅𝑎=𝑓𝑦−𝑏∪𝜇∩𝜆∪𝜇_𝑅()𝑦∪𝑓𝑥−𝑎∩𝜆∩𝜆_𝑅𝑎⊆𝑓𝑦−𝑏∩𝜆∩𝜆∪𝜇_𝑅()∪𝑓𝑥−𝑎∪𝜇∩𝜆_𝑅=𝑓𝑦−𝑏∪𝜇∩𝜆∪𝜇_𝑅(𝑥−𝑎)∩𝜆∪𝜇∪𝑓_𝑅=𝑓𝑦−𝑏∩𝜆∪𝜇_𝑅(0)∩𝜆∪𝜇. (17) According to Lemma 11, we get that (𝑓_𝑅(0)∩𝜆)∪𝜇⊆(𝑓_𝑅(𝑥𝑦−𝑎𝑏)∩𝜆)∪𝜇. Therefore, (𝑓_𝑅(𝑥𝑦−𝑎𝑏)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇; that is, (𝑥𝑦)⊕𝑓_𝑅=(𝑎𝑏)⊕𝑓_𝑅.

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

Proposition 26. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal over 𝑈. Then, (𝑅/𝑓_𝑅,+,⋅) is a ring, where 𝑅/𝑓_𝑅≜{𝑎⊕𝑓_𝑅∣𝑎∈𝑅}, (𝑥⊕𝑓_𝑅)+(𝑦⊕𝑓_𝑅)≜(𝑥+𝑦)⊕𝑓_𝑅, and (𝑥⊕𝑓_𝑅)⋅(𝑦⊕𝑓_𝑅)≜(𝑥𝑦)⊕𝑓_𝑅, for all 𝑥,𝑦∈𝑅.

Proof. It is straightforward.

Definition 27. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal over 𝑈. Then, (𝑅/𝑓_𝑅,+,⋅) is called a soft quotient ring.

Theorem 28. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal over 𝑈. Then, 𝑅/𝑓^∗_𝑅≅𝑅/𝑓_𝑅.

Proof. Assume that 𝑔∶𝑅→𝑅/𝑓_𝑅 such that 𝑔(𝑥)=𝑥⊕𝑓_𝑅, for all 𝑥∈𝑅. It is easy to see that 𝑔 is a surjective homomorphism from 𝑅 to 𝑅/𝑓_𝑅. Since Ker(𝑔)={𝑥∈𝑅∣𝑔(𝑥) = 0⊕𝑓_𝑅}={𝑥∈𝑅∣𝑥⊕𝑓_𝑅=0⊕𝑓_𝑅}={𝑥∈𝑅∣(𝑓_𝑅(𝑥)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇}=𝑓^∗_𝑅, therefore 𝑅/𝑓^∗_𝑅≅𝑅/𝑓_𝑅.

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 (𝜆,𝜇)-soft union rings related to invariant soft sets are discussed.

Definition 29. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal over 𝑈 and 𝑔_𝑅 a soft set. Then, a quotient soft subset 𝑔_𝑅/𝑓_𝑅 of 𝑅/𝑓_𝑅 is defined by 𝑔_𝑅/𝑓_𝑅𝑟⊕𝑓_𝑅=∩𝑔_𝑅(𝑥)∣𝑥∈𝑅,𝑥⊕𝑓_𝑅=𝑟⊕𝑓_𝑅, (18) for all 𝑟∈𝑅.

We will illustrate the above concept by the following example.

Example 30. Consider the ({1,2,4},{1,2})-soft union ideal 𝑓_𝑅 over ℤ⁺ in Example 15. We construct a soft set 𝑔_𝑅 over 𝑈 by 𝑔_𝑅(0)={1,2,3,5,6}, 𝑔_𝑅(1)={1,3,5,6}, 𝑔_𝑅(2)={1,2,4,5}, and 𝑔_𝑅(3)={3,6}. We define a soft set 𝑔_𝑅/𝑓_𝑅 such that 𝑔_𝑅/𝑓_𝑅(0⊕𝑓_𝑅)=𝑔_𝑅/𝑓_𝑅(2⊕𝑓_𝑅)={1,2,5}, 𝑔_𝑅/𝑓_𝑅(1⊕𝑓_𝑅)=𝑔_𝑅/𝑓_𝑅(3⊕𝑓_𝑅)={3,6}. Then, one can show that 𝑔_𝑅/𝑓_𝑅 is a quotient soft subset of 𝑅/𝑓_𝑅.

Proposition 31. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal of 𝑅 over 𝑈. Then, 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅/𝑓_𝑅, where 𝑓_𝑅(𝑥⊕𝑓_𝑅)≜𝑓_𝑅(𝑥), for all 𝑥∈𝑅.

Proof. It is easy to see that 𝑓_𝑅 is a soft set. For all 𝑥,𝑦∈𝑅, we have 𝑓_𝑅𝑥⊕𝑓_𝑅−𝑦⊕𝑓_𝑅=𝑓∩𝜆_𝑅𝑥−𝑦⊕𝑓_𝑅∩𝜆=𝑓_𝑅𝑥−𝑦∩𝜆⊆𝑓_𝑅(𝑥)∪𝑓_𝑅𝑦=𝑓∪𝜇_𝑅𝑥⊕𝑓_𝑅∪𝑓_𝑅𝑦⊕𝑓_𝑅∪𝜇. (19)
Moreover, since 𝑓_𝑅((𝑥⊕𝑓_𝑅)⋅(𝑦⊕𝑓_𝑅))∩𝜆=𝑓_𝑅((𝑥𝑦)⊕𝑓_𝑅)∩𝜆=𝑓_𝑅(𝑥𝑦)∩𝜆⊆(𝑓_𝑅(𝑥)∩𝑓_𝑅(𝑦))∪𝜇 = (𝑓_𝑅(𝑥⊕𝑓_𝑅)∩𝑓_𝑅(𝑦⊕𝑓_𝑅))∪𝜇, for all 𝑥,𝑦∈𝑅, then 𝑓_𝑅 is a (𝜆,𝜇)-soft union ideal of 𝑅/𝑓_𝑅.

Proposition 32. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal and 𝑔_𝑅 a soft union ring. Then, 𝑔_𝑅/𝑓_𝑅 is a soft union ring of 𝑅/𝑓_𝑅.

Proof. For all 𝑥,𝑦∈𝑅, we have 𝑔_𝑅/𝑓_𝑅𝑥⊕𝑓_𝑅−𝑦⊕𝑓_𝑅=𝑔_𝑅/𝑓_𝑅𝑥−𝑦⊕𝑓_𝑅=∩𝑔_𝑅(𝑢−𝑣)∣𝑢,𝑣∈𝑅,(𝑢−𝑣)⊕𝑓_𝑅=𝑥−𝑦⊕𝑓_𝑅⊆∩𝑔_𝑅(𝑢)∪𝑔_𝑅(𝑣)∣𝑢,𝑣∈𝑅,𝑢⊕𝑓_𝑅=𝑥⊕𝑓_𝑅,𝑣⊕𝑓_𝑅=𝑦⊕𝑓_𝑅=∩𝑔_𝑅(𝑢)∣𝑢∈𝑅,𝑢⊕𝑓_𝑅=𝑥⊕𝑓_𝑅∪∩𝑔_𝑅(𝑣)∣𝑣∈𝑅,𝑣⊕𝑓_𝑅=𝑦⊕𝑓_𝑅=𝑔_𝑅/𝑓_𝑅𝑥⊕𝑓_𝑅∪𝑔_𝑅/𝑓_𝑅𝑦⊕𝑓_𝑅. (20)
Similarly, we can show that 𝑔_𝑅/𝑓_𝑅((𝑥⊕𝑓_𝑅)⋅(𝑦⊕𝑓_𝑅))⊆𝑔_𝑅/𝑓_𝑅(𝑥⊕𝑓_𝑅)∪𝑔_𝑅/𝑓_𝑅(𝑦⊕𝑓_𝑅), for all 𝑥,𝑦∈𝑅. Thus, 𝑔_𝑅/𝑓_𝑅 is a soft union ring of 𝑅/𝑓_𝑅.

Definition 33. Let 𝑓_𝑅 be a (𝜆,𝜇)-soft union ideal and 𝑔_𝑅 a soft union ring. Then, (𝑔_𝑅/𝑓_𝑅,+,⋅) is called a quotient soft union ring of 𝑅/𝑓_𝑅.

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

Definition 34. Let Ψ∶𝑅₁→𝑅₂ be a ring homomorphism. A soft set 𝑓_𝑅1 over 𝑈 is called an invariant soft set with respect to Ψ if Ψ(𝑥₁)=Ψ(𝑥₂) implies 𝑓_𝑅1(𝑥₁)=𝑓_𝑅1(𝑥₂), for all 𝑥₁,𝑥₂∈𝑅₁.

Proposition 35. Let Ψ∶𝑅₁→𝑅₂ be a ring homomorphism and 𝑓_𝑅2 a soft set of 𝑅₂ over 𝑈. Then, Ψ⁻¹(𝑓_𝑅2) is an invariant soft set with respect to Ψ.

Proof. Let 𝑥₁,𝑥₂∈𝑅₁ such that Ψ(𝑥₁)=Ψ(𝑥₂). Then, Ψ⁻¹(𝑓_𝑅2)(𝑥₁)=𝑓_𝑅2(Ψ(𝑥₁)) = 𝑓_𝑅2(Ψ(𝑥₂))=Ψ⁻¹(𝑓_𝑅2)(𝑥₂). Hence, Ψ⁻¹(𝑓_𝑅2) is an invariant soft set with respect to Ψ.

Next, we establish isomorphism theorems of (𝜆,𝜇)-soft union rings.

Theorem 36 (first isomorphism theorem). Let Ψ∶𝑅₁→𝑅₂ be an epimorphism and let (𝜆,𝜇)-soft union ideal 𝑓_𝑅1 be a invariant soft set with respect to Ψ. Then, 𝑅₁/𝑓_𝑅1≅𝑅₂/Ψ(𝑓_𝑅1).

Proof. Let 𝜓∶𝑅₁→𝑅₂/Ψ(𝑓_𝑅1) be a mapping such that 𝜓(𝑥)=Ψ(𝑥)⊕Ψ(𝑓_𝑅1), for all 𝑥∈𝑅₁. Obviously, 𝜓 is an epimorphism. Since 𝑓_𝑅1 is an invariant soft set with respect to Ψ, then Ker(𝜓)={𝑥∈𝑅₁∣𝜓(𝑥)=0₂⊕Ψ(𝑓_𝑅1)}={𝑥∈𝑅₁∣Ψ(𝑥)⊕Ψ(𝑓_𝑅1)=0₂⊕Ψ(𝑓_𝑅1)}={𝑥∈𝑅₁∣(Ψ(𝑓_𝑅1)(Ψ(𝑥))∩𝜆)∪𝜇=(Ψ(𝑓_𝑅1)(0₂)∩𝜆)∪𝜇}={𝑥∈𝑅₁∣𝑥⊕𝑓_𝑅1=0₁⊕𝑓_𝑅1}=𝑓^∗_𝑅1. Therefore, 𝑅₁/𝑓^∗_𝑅1≅𝑅₂/Ψ(𝑓_𝑅1). By Theorem 28, we have 𝑅₁/𝑓^∗_𝑅1≅𝑅₁/𝑓_𝑅1. Hence, 𝑅₁/𝑓_𝑅1≅𝑅₂/Ψ(𝑓_𝑅1).

Proposition 37. Let Ψ∶𝑅₁→𝑅₂ be an epimorphism and (𝜆,𝜇)-soft union ideal and 𝑓_𝑅2 a invariant soft set with respect to Ψ. Then, 𝑅₁/Ψ⁻¹(𝑓_𝑅2)≅𝑅₂/𝑓_𝑅2.

Proof. It follows from Theorem 28 and Proposition 35 that Ψ⁻¹(𝑓_𝑅2) is a (𝜆,𝜇)-soft union ideal of 𝑅₁ and Ψ⁻¹(𝑓_𝑅2) is an invariant soft set with respect to Ψ. Since Ψ is an epimorphism, then Ψ(Ψ⁻¹(𝑓_𝑅2))=𝑓_𝑅2. By Theorem 36, we get that 𝑅₁/Ψ⁻¹(𝑓_𝑅2)≅𝑅₂/𝑓_𝑅2.

Theorem 38. Let Ψ∶𝑅₁→𝑅₂ be an epimorphism, 𝑔_𝑅1 a soft union ring of 𝑅₁, and 𝑓_𝑅1 a (𝜆,𝜇)-soft union ideal of 𝑅₁. If 𝑓_𝑅1 is an invariant soft set with respect to Ψ, then 𝑔_𝑅1/𝑓_𝑅1≅Ψ(𝑔_𝑅1)/Ψ(𝑓_𝑅1).

Proof. Let 𝜑∶𝑅₁/𝑓_𝑅1→𝑅₂/Ψ(𝑓_𝑅1) be a mapping such that 𝜑(𝑥⊕𝑓_𝑅1)=Ψ(𝑥)⊕Ψ(𝑓_𝑅1), for all 𝑥∈𝑅₁. By Theorem 36, we have that 𝜑 is a ring isomorphism. Let 𝑦∈𝑅₂. Since Ψ is a subjective homomorphism, then there exists 𝑥∈𝑅₁ such that Ψ(𝑥)=𝑦. Considering that 𝑓_𝑅1 is an invariant soft set with respect to Ψ, we obtain Ψ𝑔_𝑅1/Ψ𝑓_𝑅1𝑦⊕Ψ𝑓_𝑅1=Ψ𝑔_𝑅1/Ψ𝑓_𝑅1Ψ(𝑥)⊕Ψ𝑓_𝑅1=∩Ψ𝑔_𝑅1(𝑣)∣𝑣∈𝑅₂,𝑣⊕Ψ𝑓_𝑅1=Ψ(𝑥)⊕Ψ𝑓_𝑅1=∩∩𝑔_𝑅1(𝑢)∣𝑢∈𝑅₁,Ψ(𝑢)=𝑣∣𝑣∈𝑅₂,𝑣⊕Ψ𝑓_𝑅1=Ψ(𝑥)⊕Ψ𝑓_𝑅1=∩𝑔_𝑅1(𝑢)∣𝑢∈𝑅₁,Ψ(𝑢)⊕Ψ𝑓_𝑅1=Ψ(𝑥)⊕Ψ𝑓_𝑅1=∩𝑔_𝑅1(𝑢)∣𝑢∈𝑅₁,Ψ𝑓_𝑅1(Ψ(𝑢)(𝑥=Ψ𝑓−Ψ))∩𝜆∪𝜇_𝑅10₂𝑔∩𝜆∪𝜇=∩_𝑅1(𝑢)∣𝑢∈𝑅₁,𝑓_𝑅1()=𝑓𝑢−𝑥∩𝜆∪𝜇_𝑅10₁𝑔∩𝜆∪𝜇=∩_𝑅1(𝑢)∣𝑢∈𝑅₁,𝑢⊕𝑓_𝑅1=𝑥⊕𝑓_𝑅1=𝑔_𝑅1/𝑓_𝑅1𝑥⊕𝑓_𝑅1=𝜑⁻¹𝜑𝑔_𝑅1/𝑓_𝑅1𝑥⊕𝑓_𝑅1=𝜑𝑔_𝑅1/𝑓_𝑅1𝜑𝑥⊕𝑓_𝑅1=𝜑𝑔_𝑅1/𝑓_𝑅1Ψ(𝑥)⊕Ψ𝑓_𝑅1=𝜑𝑔_𝑅1/𝑓_𝑅1𝑦⊕Ψ𝑓_𝑅1. (21) Hence, 𝜑(𝑔_𝑅1/𝑓_𝑅1)=Ψ(𝑔_𝑅1)/Ψ(𝑓_𝑅1), and so 𝑔_𝑅1/𝑓_𝑅1≅Ψ(𝑔_𝑅1)/Ψ(𝑓_𝑅1).

Proposition 39. Let Ψ∶𝑅₁→𝑅₂ be an epimorphism. If 𝑔_𝑅2 is a soft union ring of 𝑅₂ and 𝑓_𝑅2 is a (𝜆,𝜇)-soft union ideal of 𝑅₂, then Ψ⁻¹(𝑔_𝑅2)/Ψ⁻¹(𝑓_𝑅2)≅𝑔_𝑅2/𝑓_𝑅2.

Proof. Since Ψ is a subjective homomorphism, then Ψ(Ψ⁻¹(𝑓_𝑅2))=𝑓_𝑅2 and Ψ(Ψ⁻¹(𝑔_𝑅2))=𝑔_𝑅2. By Proposition 35, we get that Ψ⁻¹(𝑓_𝑅1) is an invariant soft set with respect to Ψ. It follows from Theorem 38 that Ψ⁻¹(𝑔_𝑅2)/Ψ⁻¹(𝑓_𝑅2)≅𝑔_𝑅2/𝑓_𝑅2.

Lemma 40. Let 𝑓_𝑅 and 𝑔_𝑅 be (𝜆,𝜇)-soft union ideals of 𝑅 such that 𝑔_𝑅⊂𝑓_𝑅 and (𝑓_𝑅(0)∩𝜆)∪𝜇=(𝑔_𝑅(0)∩𝜆)∪𝜇. Then, 𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅 if and only if 𝑥⊕𝑔_𝑅=𝑦⊕𝑔_𝑅, for all 𝑥,𝑦∈𝑅.

Proof. Assume that 𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅, for all 𝑥,𝑦∈𝑅. Then, we have that ((𝑔_𝑅/𝑓_𝑅((𝑥−𝑦)⊕𝑓_𝑅))∩𝜆)∪𝜇=((𝑔_𝑅/𝑓_𝑅(0⊕𝑓_𝑅))∩𝜆)∪𝜇=(𝑔_𝑅(0)∩𝜆)∪𝜇, and so ∩𝑔_𝑅(𝑡)∣𝑡∈𝑅,𝑡⊕𝑓_𝑅=𝑥−𝑦⊕𝑓_𝑅𝑔∩𝜆∪𝜇=∩_𝑅(𝑡)∩𝜆∪𝜇∣𝑡∈𝑅,𝑡⊕𝑓_𝑅=𝑥−𝑦⊕𝑓_𝑅=𝑔_𝑅(0)∩𝜆∪𝜇. (22)
If 𝑡⊕𝑓_𝑅=(𝑥−𝑦)⊕𝑓_𝑅, then (𝑓_𝑅(𝑥−𝑦−𝑡)∩𝜆)∪𝜇=(𝑓_𝑅(0)∩𝜆)∪𝜇. Therefore, 𝑔_𝑅=𝑔𝑥−𝑦∩𝜆∪𝜇_𝑅=𝑔𝑥−𝑦−𝑡+𝑡∩𝜆∪𝜇_𝑅⊆𝑔𝑥−𝑦−𝑡+𝑡∩𝜆∩𝜆∪𝜇_𝑅𝑥−𝑦−𝑡∪𝑔_𝑅(𝑡)=𝑔∪𝜇∩𝜆∪𝜇_𝑅𝑥−𝑦−𝑡∩𝜆∪𝜇∪𝑔_𝑅(𝑡)⊆𝑓∩𝜆∪𝜇_𝑅𝑥−𝑦−𝑡∩𝜆∪𝜇∪𝑔_𝑅(𝑡)=𝑓∩𝜆∪𝜇_𝑅(0)∩𝜆∪𝜇∪𝑔_𝑅(𝑡)⊆𝑔∩𝜆∪𝜇_𝑅(𝑡)∩𝜆∪𝜇. (23) It follows that (𝑔_𝑅(𝑥−𝑦)∩𝜆)∪𝜇⊆∩{(𝑔_𝑅(𝑡)∩𝜆)∪𝜇∣𝑡∈𝑅,𝑡⊕𝑓_𝑅=(𝑥−𝑦)⊕𝑓_𝑅} = (𝑔_𝑅(0)∩𝜆)∪𝜇. On the other hand, we have (𝑔_𝑅(0)∩𝜆)∪𝜇⊆(𝑔_𝑅(𝑥−𝑦)∩𝜆)∪𝜇, by Lemma 11. Hence, (𝑔_𝑅(𝑥−𝑦)∩𝜆)∪𝜇=(𝑔_𝑅(0)∩𝜆)∪𝜇. And thus, 𝑥⊕𝑔_𝑅=𝑦⊕𝑔_𝑅.
Conversely, suppose that 𝑥⊕𝑔_𝑅=𝑦⊕𝑔_𝑅, for all 𝑥,𝑦∈𝑅. It follows that (𝑔_𝑅(𝑥−𝑦)∩𝜆)∪𝜇=(𝑔_𝑅(0)∩𝜆)∪𝜇=(𝑔_𝑅/𝑓_𝑅(0⊕𝑓_𝑅)∩𝜆)∪𝜇. Therefore, 𝑔_𝑅𝑥−𝑦⊕𝑓_𝑅=∩𝑔∩𝜆∪𝜇_𝑅(𝑡)∣𝑡∈𝑅,𝑡⊕𝑓_𝑅=𝑥−𝑦⊕𝑓_𝑅𝑔∩𝜆∪𝜇=∩_𝑅(𝑡)∩𝜆∪𝜇∣𝑡∈𝑅,𝑡⊕𝑓_𝑅=𝑥−𝑦⊕𝑓_𝑅=𝑔_𝑅(0)=𝑔∩𝜆∪𝜇_𝑅/𝑓_𝑅0⊕𝑓_𝑅∩𝜆∪𝜇; (24) that is, (𝑔_𝑅/𝑓_𝑅((𝑥⊕𝑓_𝑅)−(𝑦⊕𝑓_𝑅))∩𝜆)∪𝜇=(𝑔_𝑅/𝑓_𝑅(0⊕𝑓_𝑅)∩𝜆)∪𝜇. It follows from Proposition 24 that 𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅.

Theorem 41 (third isomorphism theorem). Let 𝑓_𝑅 and 𝑔_𝑅 be (𝜆,𝜇)-soft union ideals of 𝑅 such that 𝑔_𝑅⊆𝑓_𝑅 and (𝑓_𝑅(0)∩𝜆)∪𝜇=(𝑔_𝑅(0)∩𝜆)∪𝜇. If ℎ_𝑅 is a soft union ring, then (ℎ_𝑅/𝑓_𝑅)/(𝑔_𝑅/𝑓_𝑅)≅ℎ_𝑅/𝑔_𝑅.

Proof. Let Ψ∶(𝑅/𝑓_𝑅)/(𝑅/𝑓_𝑅)→𝑅/𝑔_𝑅 be a mapping such that Ψ(𝑟⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅)=𝑟⊕𝑔_𝑅, for all 𝑟∈𝑅. According to Lemma 40, we obtain that Ψ is an injective mapping. Obviously, Ψ is subjective. For all 𝑥,𝑦∈𝑅, we have Ψ𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅+𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=Ψ𝑥+𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑥+𝑦⊕𝑔_𝑅=𝑥⊕𝑔_𝑅+𝑦⊕𝑔_𝑅,Ψ𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅⋅𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=Ψ𝑥𝑦⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑥𝑦⊕𝑔_𝑅=𝑥⊕𝑔_𝑅⋅𝑦⊕𝑔_𝑅. (25) Thus, Ψ is a ring isomorphism.
Moreover, since ℎ_𝑅/𝑓_𝑅/𝑔_𝑅/𝑓_𝑅𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=∩ℎ_𝑅/𝑓_𝑅𝑢⊕𝑓_𝑅∣𝑢∈𝑅,𝑢⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=∩∩ℎ_𝑅(𝑣)∣𝑣∈𝑅,𝑣⊕𝑓_𝑅=𝑢⊕𝑓_𝑅∣𝑢∈𝑅,𝑢⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=∩ℎ_𝑅(𝑣)∣𝑣∈𝑅,𝑣⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=∩ℎ_𝑅(𝑣)∣𝑣∈𝑅,𝑣⊕𝑔_𝑅=𝑥⊕𝑔_𝑅=ℎ_𝑅/𝑔_𝑅𝑥⊕𝑔_𝑅=ℎ_𝑅/𝑔_𝑅Ψ𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅=Ψ⁻¹ℎ_𝑅/𝑔_𝑅𝑥⊕𝑓_𝑅⊕𝑔_𝑅/𝑓_𝑅, (26) that is, (ℎ_𝑅/𝑓_𝑅)/(𝑔_𝑅/𝑓_𝑅)=Ψ⁻¹(ℎ_𝑅/𝑔_𝑅) therefore Ψ((ℎ_𝑅/𝑓_𝑅)/(𝑔_𝑅/𝑓_𝑅))=ℎ_𝑅/𝑔_𝑅, and so (ℎ_𝑅/𝑓_𝑅)/(𝑔_𝑅/𝑓_𝑅)≅ℎ_𝑅/𝑔_𝑅.

6. Conclusions

We introduced the notions of (𝜆,𝜇)-soft union rings and (𝜆,𝜇)-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 (𝜆,𝜇)-soft union ideals by using soft cosets and gave an approach for constructing quotient soft union rings. Finally, we considered isomorphism theorems of (𝜆,𝜇)-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 (𝜆,𝜇)-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).