# Applications of Soft Union Sets in the Ring Theory

Authors: Yongwei Yang; Xiaolong Xin; Pengfei He
Journal: Journal of Applied Mathematics
 (2013)

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## 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 ofλ,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.


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## Body

## 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 subgroupS 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 setfA over U is defined as fA:E→𝒫(U) such that fA(x)=∅ if x∉A.In other words, a soft set overU is a parameterized family of subsets of the universe U. For all ε∈A, fA(ε) may be considered as the set of ε-approximate elements of the soft set fA. A soft set fA over U can be presented by the set of ordered pairs:
(1)fA={(x,fA(x))∣x∈E,fA(x)∈𝒫(U)}.
Clearly, a soft set is not a set. For illustration, Molodtsov considered several examples in [3].IffA is a soft set over U, then the image of fA is defined by Im(fA)={fA(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]).
LetfA,fB∈S(U). If fA(x)⊆fB(x), for all x∈E, then fA is called a soft subset of fB and denoted by fA⊆~fB.
f
A and fB are called soft equal, denoted by fA=fB, if and only if fA⊆~fB and fB⊆~fA.Definition 3 (see [24]).
LetfA,fB∈S(U) and let Ψ be a function from A to B. Then, the soft anti-image of fA under Ψ, denoted by Ψ(fA), is a soft set over U defined by
(2)Ψ(fA)(b)={∩{fA(a)∣a∈A,Ψ(a)=b},ifΨ-1(b)≠∅,∅,otherwise,
for all b∈B. And the soft preimage of fB under Ψ, denoted by Ψ-1(fB), is a soft set over U defined by Ψ-1(fB)(a)=fB(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]).
LetfA be a soft set over U and α⊆U. Then, lower α-inclusion of fA, denoted by L(fA;α), is defined as L(fA;α)={x∈A∣fA(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 setfR over U is called a soft union ring of R if (1)
fR(x+y)⊆fR(x)∪fR(y),(2)
fR(x)⊆fR(-x),(3)
fR(xy)⊆fR(x)∪fR(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 setfR over U is called a soft union left (resp., right) ideal of R if (1)
fR(x-y)⊆fR(x)∪fR(y),(2)
fR(xy)⊆fR(y)  (resp., fR(xy)⊆fR(x)),for all x,y∈R.
A soft setfR 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.
LetfR be a soft set over U. fR is called a (λ,μ)-soft union ring of R if (1)
fR(x+y)∩λ⊆fR(x)∪fR(y)∪μ,(2)
fR(x)∩λ⊆fR(-x)∪μ,(3)
fR(xy)∩λ⊆fR(x)∪fR(y)∪μ,for all x,y∈R.Remark 8.
LetfR be a soft union ring of R over U; then fR 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 ringR={[ab00]∣a,b∈ℤ2} with the operations addition and multiplication of matrices, U=ℤ5. We define a soft set fR over U by
(3)fR([0000])={0,2},fR([0100])={0,1,2,4},fR([1000])={0,1,2},fR([1100])={0,1,4}.
Then, one can easily show that fR is a ({0,1,2,4},{0,2})-soft union ring of R. But fR is not a soft union ring of R because
(4)fR([1100]+[1100])=fR([0000])={0,2}⊈fR([1100])∪fR([1100])={0,1,4}.In order to give some characterizations of(λ,μ)-soft union rings, we need the following lemmas.Lemma 10.
LetfR be a soft set over U. If fR is a (λ,μ)-soft union ring of R, then (1)
(fR(-x)∩λ)∪μ⊆(fR(x)∩λ)∪μ,(2)
fR(-x)∩λ⊆fR(x)∪μ,for all x∈R.Proof.
(
1
) Assume that fR is a (λ,μ)-soft union ring of R; then for all x∈R, we get that  (fR(-x)∩λ)∪μ=(fR(-x)∩λ∩λ)∪μ⊆((fR(-(-x))∪μ)∩λ)∪μ=(fR(x)∩λ)∪μ.
(
2
) It is straightforward.Lemma 11.
LetfR be a soft set over U. If fR is a (λ,μ)-soft union ring of R, then (1)
(fR(0)∩λ)∪μ⊆(fR(x)∩λ)∪μ,(2)
fR(0)∩λ⊆fR(x)∪μ,for all x∈R.Proof.
(
1
) Assume that fR is a (λ,μ)-soft union ring of R; then for all x∈R, we have that (fR(0)∩λ)∪μ=(fR(x-x)∩λ∩λ)∪μ⊆((fR(x)∪fR(-x)∪μ)∩λ)∪μ=((fR(x)∩λ)∪μ)∪(fR(-x)∩λ)∪μ⊆(fR(x)∩λ)∪μ, by Lemma 10.
(
2
) It is straightforward.Combining Lemma11 and Definition 7, we obtain the following characterization of (λ,μ)-soft union rings.Theorem 12.
LetfR be a soft set over U. Then, fR is a (λ,μ)-soft union ring of R if and only if fR satisfies the following conditions: (1)
fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ,(2)
fR(xy)∩λ⊆fR(x)∪fR(y)∪μ,for all x,y∈R.Proof.
Assume thatfR is a (λ,μ)-soft union ring of R. By Definition 7 and Lemma 11, we have fR(xy)∩λ⊆fR(x)∪fR(y)∪μ and
(5)fR(x-y)∩λ=(fR(x-y)∩λ)∩λ⊆(fR(x)∪fR(-y)∪μ)∩λ=(fR(x)∩λ)∪(fR(-y)∩λ)∪μ⊆fR(x)∪fR(y)∪μ,
for all x,y∈R.
Conversely, sincefR(x-y)∩λ⊆(fR(x)∪fR(y))∪μ, for all x,y∈R, by Lemma 11, we get
(6)fR(x)∩λ=(fR(0-(-x))∩λ)∩λ⊆(fR(0)∪fR(-x)∪μ)∩λ=(fR(0)∩λ)∪((fR(-x)∪μ)∩λ)⊆(fR(-x)∪μ)∪(fR(-x)∪μ)=fR(-x)∪μ.
Moreover, we have
(7)fR(x+y)∩λ=(fR(x-(-y))∩λ)∩λ⊆(fR(x)∪fR(-y)∪μ)∩λ=((fR(x)∪μ)∩λ)∪(fR(-y)∩λ)⊆fR(x)∪fR(y)∪μ.
And it follows from hypothesis that fR(xy)∩λ⊆fR(x)∪fR(y)∪μ; therefore fR 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 setfR over U is called a (λ,μ)-soft union ideal of R if (1)
fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ,(2)
fR(xy)∩λ⊆(fR(x)∩fR(y))∪μ,for all x,y∈R.Remark 14.
LetfR be a soft union ideal of R over U; then fR 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.
LetR=ℤ4 be a ring and U=ℤ+ a universal set. We construct a soft set fR over U by fR(0)={1}, fR(1)=fR(3)={2,3,4}, and fR(2)={3}. One can show that fR is a ({1,2,4},{1,2})-soft union ideal. But fR is not a soft union ideal of R, since fR(2-2)=fR(0)={1}⊈fR(2)∪fR(2)={3}.Proposition 16.
IffR is a (λ,μ)-soft union ideal of R, then fR*≜{x∈R∣(fR(x)∩λ)∪μ=(fR(0)∩λ)∪μ} is an ideal of R.Proof.
We need to show that (i)x-y∈fR*, (ii) rx∈fR*, and (iii) xr∈fR*, for all x,y∈fR* and r∈R. If x,y∈fR*, then (fR(x)∩λ)∪μ=(fR(y)∩λ)∪μ=(fR(0)∩λ)∪μ. By Lemma 11, we obtain that (fR(0)∩λ)∪μ⊆(fR(x-y)∩λ)∪μ, (fR(0)∩λ)∪μ⊆(fR(rx)∩λ)∪μ, and (fR(0)∩λ)∪μ⊆(fR(xr)∩λ)∪μ, for all x,y∈fR* and r∈R. Since fR is a (λ,μ)-soft union ideal, then for all x,y∈fR* and r∈R, we have
(8)(fR(x-y)∩λ)∪μ=(fR(x-y)∩λ∩λ)∪μ⊆((fR(x)∪fR(y)∪μ)∩λ)∪μ=((fR(x)∩λ)∪μ)∪((fR(y)∩λ)∪μ)=(fR(0)∩λ)∪μ,(fR(rx)∩λ)∪μ=(fR(rx)∩λ∩λ)∪μ⊆(((fR(r)∩fR(x))∪μ)∩λ)∪μ⊆(fR(x)∩λ)∪μ=(fR(0)∩λ)∪μ.
Similarly, we can prove that (fR(xr)∩λ)∪μ⊆(fR(0)∩λ)∪μ, for all x∈fR* and r∈R. Hence, (fR(x-y)∩λ)∪μ=(fR(rx)∩λ)∪μ=(fR(xr)∩λ)∪μ=(fR(0)∩λ)∪μ. Therefore, (i) x-y∈fR*, (ii) rx∈fR*, and (iii)   xr∈fR*, for all x,y∈fR* and r∈R. And thus, fR* 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.
LetfR be a (λ,μ)-soft union ideal of R; then EIm(fR) is called the extended image set of fR, where EIm(fR)=Im(fR)∪{λ,μ}.Now, we characterize(λ,μ)-soft union ideals by lower inclusions.Proposition 18.
LetfR be a soft set over U and EIm(fR) a totally ordered set by inclusion. Then, fR is a (λ,μ)-soft union ideal of R if and only if L(fR;α) is an ideal of R, whenever it is nonempty, for each α⊆U where μ⊆α⊂λ.Proof.
Assume thatfR is a (λ,μ)-soft union ideal of R and L(fR;α) is nonempty. It is sufficient to show that x-y∈L(fR;α), rx∈L(fR;α) and xr∈L(fR;α), for all x,y∈L(fR;α), and r∈R. Let x,y∈L(fR;α). It follows that fR(x)⊆α and fR(y)⊆α. Since fR is a (λ,μ)-soft union ideal of R and EIm(fR) is a totally ordered set, then fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ⊆α∪α∪α=α⊂λ, fR(rx)∩λ⊆(fR(r)∩fR(x))∪μ⊆α∪α=α⊂λ, and fR(xr)∩λ⊆(fR(x)∩fR(r))∪μ⊆α∪α=α⊂λ. And thus, fR(x-y)⊆α, fR(rx)⊆α, and fR(xr)⊆α; hence x-y,rx,xr∈L(fR;α). Therefore, L(fR;α) is an ideal of R.
Conversely, assume thatL(fR;α) is an ideal of R whenever it is nonempty, for each α⊆U where μ⊆α⊂λ. Suppose that fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ does not hold for some x,y∈R; then there exist x0,y0∈R such that fR(x0-y0)∩λ⊃α=(fR(x0)∪fR(y0)∪μ). Therefore, fR(x0)∪fR(y0)⊆α and μ⊆α⊂λ. It follows that x0,y0∈L(fR;α), but fR(x0-y0)⊃α; that is, x0-y0∉L(fR;α), which is a contradiction. Hence, fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ, for all x,y∈R. Similarly, we can prove that fR(xy)∩λ⊆(fR(x)∩fR(y))∪μ, for all x,y∈R. Thus, fR 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.
LetfR1 be a soft set over U and Ψ a ring epimorphism from R1 to R2. If fR1 is a (λ,μ)-soft union ideal of R1, then Ψ(fR1) is a (λ,μ)-soft union ideal of R2 and (Ψ(fR1)(02)∩λ)∪μ=(fR1(01)∩λ)∪μ.Proof.
Lety1,y2∈R2 and fR1 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
(9)Ψ(fR1)(y1-y2)∩λ=∩{fR1(x1-x2)∣Ψ(x1-x2)=y1-y2}∩λ=∩{fR1(x1-x2)∩λ∣Ψ(x1-x2)=y1-y2}⊆∩{fR1(x1)∪fR1(x2)∪μ∣Ψ(x1)=y1,Ψ(x2)=y2}=∩{fR1(x1)∣Ψ(x1)=y1}∪∩{fR1(x2)∣Ψ(x2)=y2}∪μ=Ψ(fR1)(y1)∪Ψ(fR1)(y2)∪μ,Ψ(fR1)(y1y2)∩λ=∩{fR1(x1x2)∣Ψ(x1x2)=y1y2}∩λ=∩{fR1(x1x2)∩λ∣Ψ(x1x2)=y1y2}⊆∩{(fR1(x1)∩fR1(x2))∪μ∣Ψ(x1)=y1,Ψ(x2)=y2}=(∩{fR1(x1)∣Ψ(x1)=y1}∩∩∩∩{fR1(x2)∣Ψ(x2)=y2})∪μ=(Ψ(fR1)(y1)∩Ψ(fR1)(y1))∪μ.
Therefore, Ψ(fR1) is a (λ,μ)-soft union ideal of R2.
By Lemma11, we have (Ψ(fR1)(02)∩λ)∪μ=(∩{fR1(x)∣x∈R1,Ψ(x)=02}∩λ)∪μ = ∩{(fR1(x)∩λ)∪μ∣x∈R1,Ψ(x)=02}=(fR1(01)∩λ)∪μ.Theorem 20.
LetfR2 be a soft set over U and Ψ a ring homomorphism from R1 to R2. If fR2 is a (λ,μ)-soft union ideal of R2, then Ψ-1(fR2) is a (λ,μ)-soft union ideal of R1.Proof.
Letx1,x2∈R1. Then,
(10)Ψ-1(fR2)(x1-x2)∩λ=fR2(Ψ(x1-x2))∩λ=fR2(Ψ(x1)-Ψ(x2))∩λ⊆fR2(Ψ(x1))∪fR2(Ψ(x2))∪μ=Ψ-1(fR2)(x1)∪Ψ-1(fR2)(x1)∪μ.
Moreover, we have(11)Ψ-1(fR2)(x1x2)∩λ=fR2(Ψ(x1x2))∩λ=fR2(Ψ(x1)·Ψ(x2))∩λ⊆(fR2(Ψ(x1))∩fR2(Ψ(x1)))∪μ=(Ψ-1(fR2)(x1)∩Ψ-1(fR2)(x1))∪μ.
Hence,Ψ-1(fR2) 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.
LetfR be a (λ,μ)-soft union ring of R over U and r∈R. Then, a soft coset r⊕fR of fR is defined by
(12)(r⊕fR)(x)=(fR(x-r)∩λ)∪μ,
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 fR in Example 9. If r=[0100], then we define a soft set r⊕fR as
(13)(r⊕fR)([0000])={0,1,2,4},(r⊕fR)([0100])={0,2},(r⊕fR)([1000])={0,1,2,4},(r⊕fR)([1100])={0,1,2}.
Then, it is easy to show that r⊕fR is a soft coset of fR.Proposition 23.
LetfR be a (λ,μ)-soft union ring over U and a,b∈R. Then, (fR(a-b)∩λ)∪μ=(fR(0)∩λ)∪μ if and only if (fR(b-a)∩λ)∪μ=(fR(0)∩λ)∪μ.Proof.
Suppose that(fR(b-a)∩λ)∪μ=(fR(0)∩λ)∪μ. Since fR is a (λ,μ)-soft union ring, then (fR(a-b)∩λ)∪μ=(fR(a-b)∩λ∩λ)∪μ⊆((fR(b-a)∪μ)∩λ)∪μ=(fR(b-a)∩λ)∪μ = (fR(0)∩λ)∪μ. By Lemma 11, we have (fR(0)∩λ)∪μ⊆(fR(a-b)∩λ)∪μ. Thus, (fR(a-b)∩λ)∪μ=(fR(0)∩λ)∪μ.
Conversely, assume that(fR(a-b)∩λ)∪μ=(fR(0)∩λ)∪μ. We can prove that (fR(b-a)∩λ)∪μ=(fR(0)∩λ)∪μ in a similar way.Proposition 24.
LetfR be a (λ,μ)-soft union ring over U and a,b∈R. Then, a⊕fR=b⊕fR if and only if (fR(a-b)∩λ)∪μ=(fR(0)∩λ)∪μ.Proof.
Suppose that(fR(a-b)∩λ)∪μ=(fR(0)∩λ)∪μ; then
(14)(b⊕fR)(x)=(fR(x-b)∩λ)∪μ=(fR(x-a+a-b)∩λ)∪μ=(fR(x-a+a-b)∩λ∩λ)∪μ⊆((fR(x-a)∪fR(a-b)∪μ)∩λ)∪μ=((fR(x-a)∩λ)∪μ)∪((fR(a-b)∩λ)∪μ)=((fR(x-a)∩λ)∪μ)∪((fR(0)∩λ)∪μ)⊆(fR(x-a)∩λ)∪μ=(a⊕fR)(x),
for all x∈R. Therefore, b⊕fR⊆~a⊕fR. Similarly, we can show that a⊕fR⊆~b⊕fR. Hence, a⊕fR=b⊕fR.
Conversely, assume thata⊕fR=b⊕fR. It follows that (fR(a-b)∩λ)∪μ=(b⊕fR)(a) = (a⊕fR)(a)=(fR(0)∩λ)∪μ.Based on the above proposition, we give a property related to soft cosets as follows.Proposition 25.
LetfR be a (λ,μ)-soft union ideal over U and a,b,x,y∈R. If
(15)x⊕fR=a⊕fR,y⊕fR=b⊕fR,then(x+y)⊕fR=(a+b)⊕fR,xy⊕fR=ab⊕fR.Proof.
Suppose thatx⊕fR=a⊕fR, y⊕fR=b⊕fR. Then, (fR(x-a)∩λ)∪μ=(fR(0)∩λ)∪μ and (fR(y-b)∩λ)∪μ=(fR(0)∩λ)∪μ, by Proposition 24. Since fR is a (λ,μ)-soft union ideal, then
(16)(fR((x+y)-(a+b))∩λ)∪μ=(fR((x-a)+(y-b))∩λ)∪μ=(fR((x-a)+(y-b))∩λ∩λ)∪μ⊆((fR(x-a)∪fR(y-b)∪μ)∩λ)∪μ=(fR(x-a)∩λ)∪(fR(y-b)∩λ)∪μ=(fR(0)∩λ)∪μ.
On the other hand, it follows from Lemma 11 that (fR(0)∩λ)∪μ⊆(fR((x+y)-(a+b))∩λ)∪μ. Hence, (fR((x+y)-(a+b))∩λ)∪μ=(fR(0)∩λ)∪μ, and so (x+y)⊕fR=(a+b)⊕fR.
Moreover,(17)(fR(xy-ab)∩λ)∪μ=(fR((x-a)y+a(y-b))∩λ)∪μ=(fR((x-a)y+a(y-b))∩λ∩λ)∪μ⊆((fR((x-a)y)∪fR(a(y-b))∪μ)∩λ)∪μ=(fR((x-a)y)∩λ∩λ)∪(fR(a(y-b))∩λ∩λ)∪μ⊆((fR(x-a)∪μ)∩λ)∪((fR(y-b)∪μ)∩λ)∪μ=((fR(x-a)∩λ)∪μ)∪((fR(y-b)∩λ)∪μ)=(fR(0)∩λ)∪μ.
According to Lemma 11, we get that (fR(0)∩λ)∪μ⊆(fR(xy-ab)∩λ)∪μ. Therefore, (fR(xy-ab)∩λ)∪μ=(fR(0)∩λ)∪μ; that is, (xy)⊕fR=(ab)⊕fR.In view of Proposition25, we have the following result.Proposition 26.
LetfR be a (λ,μ)-soft union ideal over U. Then, (R/fR,+,·) is a ring, where R/fR≜{a⊕fR∣a∈R}, (x⊕fR)+(y⊕fR)≜(x+y)⊕fR, and (x⊕fR)·(y⊕fR)≜(xy)⊕fR, for all x,y∈R.Proof.
It is straightforward.Definition 27.
LetfR be a (λ,μ)-soft union ideal over U. Then, (R/fR,+,·) is called a soft quotient ring.Theorem 28.
LetfR be a (λ,μ)-soft union ideal over U. Then, R/fR*≅R/fR.Proof.
Assume thatg:R→R/fR such that g(x)=x⊕fR, for all x∈R. It is easy to see that g is a surjective homomorphism from R to R/fR. Since Ker(g)={x∈R∣g(x) = 0⊕fR}={x∈R∣x⊕fR=0⊕fR}={x∈R∣(fR(x)∩λ)∪μ=(fR(0)∩λ)∪μ}=fR*, therefore R/fR*≅R/fR.


## 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.
LetfR be a (λ,μ)-soft union ideal over U and gR a soft set. Then, a quotient soft subset gR/fR of R/fR is defined by
(18)gR/fR(r⊕fR)=∩{gR(x)∣x∈R,x⊕fR=r⊕fR},
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 fR over ℤ+ in Example 15. We construct a soft set gR over U by gR(0)={1,2,3,5,6}, gR(1)={1,3,5,6}, gR(2)={1,2,4,5}, and gR(3)={3,6}. We define a soft set gR/fR such that gR/fR(0⊕fR)=gR/fR(2⊕fR)={1,2,5}, gR/fR(1⊕fR)=gR/fR(3⊕fR)={3,6}. Then, one can show that gR/fR is a quotient soft subset of R/fR.Proposition 31.
LetfR be a (λ,μ)-soft union ideal of R over U. Then, f^R is a (λ,μ)-soft union ideal of R/fR, where f^R(x⊕fR)≜fR(x), for all x∈R.Proof.
It is easy to see thatf^R is a soft set. For all x,y∈R, we have
(19)f^R((x⊕fR)-(y⊕fR))∩λ=f^R((x-y)⊕fR)∩λ=fR(x-y)∩λ⊆fR(x)∪fR(y)∪μ=f^R(x⊕fR)∪f^R(y⊕fR)∪μ.
Moreover, sincef^R((x⊕fR)·(y⊕fR))∩λ=f^R((xy)⊕fR)∩λ=fR(xy)∩λ⊆(fR(x)∩fR(y))∪μ = (f^R(x⊕fR)∩f^R(y⊕fR))∪μ, for all x,y∈R, then f^R is a (λ,μ)-soft union ideal of R/fR.Proposition 32.
LetfR be a (λ,μ)-soft union ideal and gR a soft union ring. Then, gR/fR is a soft union ring of R/fR.Proof.
For allx,y∈R, we have
(20)gR/fR((x⊕fR)-(y⊕fR))=gR/fR((x-y)⊕fR)=∩{gR(u-v)∣u,v∈R,(u-v)=⊕fR=(x-y)⊕fR}⊆∩{gR(u)∪gR(v)∣u,v∈R,u⊕fR=x⊕fR,v⊕fR=y⊕fR}=∩{gR(u)∣u∈R,u⊕fR=x⊕fR}∪∩{gR(v)∣v∈R,v⊕fR=y⊕fR}=gR/fR(x⊕fR)∪gR/fR(y⊕fR).
Similarly, we can show thatgR/fR((x⊕fR)·(y⊕fR))⊆gR/fR(x⊕fR)∪gR/fR(y⊕fR), for all x,y∈R. Thus, gR/fR is a soft union ring of R/fR.Definition 33.
LetfR be a (λ,μ)-soft union ideal and gR a soft union ring. Then, (gR/fR,+,·) is called a quotient soft union ring of R/fR.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.
LetΨ:R1→R2 be a ring homomorphism. A soft set fR1 over U is called an invariant soft set with respect to Ψ if Ψ(x1)=Ψ(x2) implies fR1(x1)=fR1(x2), for all x1,x2∈R1.Proposition 35.
LetΨ:R1→R2 be a ring homomorphism and fR2 a soft set of R2 over U. Then, Ψ-1(fR2) is an invariant soft set with respect to Ψ.Proof.
Letx1,x2∈R1 such that Ψ(x1)=Ψ(x2). Then, Ψ-1(fR2)(x1)=fR2(Ψ(x1)) = fR2(Ψ(x2))=Ψ-1(fR2)(x2). Hence, Ψ-1(fR2) 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).
LetΨ:R1→R2 be an epimorphism and let (λ,μ)-soft union ideal fR1 be a invariant soft set with respect to Ψ. Then, R1/fR1≅R2/Ψ(fR1).Proof.
Letψ:R1→R2/Ψ(fR1) be a mapping such that ψ(x)=Ψ(x)⊕Ψ(fR1), for all x∈R1. Obviously, ψ is an epimorphism. Since fR1 is an invariant soft set with respect to Ψ, then Ker(ψ)={x∈R1∣ψ(x)=02⊕Ψ(fR1)}={x∈R1∣Ψ(x)⊕Ψ(fR1)=02⊕Ψ(fR1)}={x∈R1∣(Ψ(fR1)(Ψ(x))∩λ)∪μ=(Ψ(fR1)(02)∩λ)∪μ}={x∈R1∣x⊕fR1=01⊕fR1}=fR1*.  Therefore, R1/fR1*≅R2/Ψ(fR1). By Theorem 28, we have R1/fR1*≅R1/fR1. Hence, R1/fR1≅R2/Ψ(fR1).Proposition 37.
LetΨ:R1→R2 be an epimorphism and (λ,μ)-soft union ideal and fR2 a invariant soft set with respect to Ψ. Then, R1/Ψ-1(fR2)≅R2/fR2.Proof.
It follows from Theorem28 and Proposition 35 that Ψ-1(fR2) is a (λ,μ)-soft union ideal of R1 and Ψ-1(fR2) is an invariant soft set with respect to Ψ. Since Ψ is an epimorphism, then Ψ(Ψ-1(fR2))=fR2. By Theorem 36, we get that R1/Ψ-1(fR2)≅R2/fR2.Theorem 38.
LetΨ:R1→R2 be an epimorphism, gR1 a soft union ring of R1, and fR1 a (λ,μ)-soft union ideal of R1. If fR1 is an invariant soft set with respect to Ψ, then gR1/fR1≅Ψ(gR1)/Ψ(fR1).Proof.
Letφ:R1/fR1→R2/Ψ(fR1) be a mapping such that φ(x⊕fR1)=Ψ(x)⊕Ψ(fR1), 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 fR1 is an invariant soft set with respect to Ψ, we obtain
(21)Ψ(gR1)/Ψ(fR1)(y⊕Ψ(fR1))=Ψ(gR1)/Ψ(fR1)(Ψ(x)⊕Ψ(fR1))=∩{Ψ(gR1)(v)∣v∈R2,v⊕Ψ(fR1)=Ψ(x)⊕Ψ(fR1)}=∩{∩{gR1(u)∣u∈R1,Ψ(u)=v}∣v∈R2,v⊕Ψ(fR1)=Ψ(x)⊕Ψ(fR1)}=∩{gR1(u)∣u∈R1,Ψ(u)⊕Ψ(fR1)=Ψ(x)⊕Ψ(fR1)}=∩{gR1(u)∣u∈R1,(Ψ(fR1)(Ψ(u)-Ψ(x))∩λ)∪μ=(Ψ(fR1)(02)∩λ)∪μ}=∩{gR1(u)∣u∈R1,(fR1(u-x)∩λ)∪μ=(fR1(01)∩λ)∪μ}=∩{gR1(u)∣u∈R1,u⊕fR1=x⊕fR1}=gR1/fR1(x⊕fR1)=φ-1(φ(gR1/fR1))(x⊕fR1)=φ(gR1/fR1)(φ(x⊕fR1))=φ(gR1/fR1)(Ψ(x)⊕Ψ(fR1))=φ(gR1/fR1)(y⊕Ψ(fR1)).
Hence, φ(gR1/fR1)=Ψ(gR1)/Ψ(fR1), and so gR1/fR1≅Ψ(gR1)/Ψ(fR1).Proposition 39.
LetΨ:R1→R2 be an epimorphism. If gR2 is a soft union ring of R2 and fR2 is a (λ,μ)-soft union ideal of R2, then Ψ-1(gR2)/Ψ-1(fR2)≅gR2/fR2.Proof.
SinceΨ is a subjective homomorphism, then Ψ(Ψ-1(fR2))=fR2 and Ψ(Ψ-1(gR2))=gR2. By Proposition 35, we get that Ψ-1(fR1) is an invariant soft set with respect to Ψ. It follows from Theorem 38 that Ψ-1(gR2)/Ψ-1(fR2)≅gR2/fR2.Lemma 40.
LetfR and gR be (λ,μ)-soft union ideals of R such that gR⊂~fR and (fR(0)∩λ)∪μ=(gR(0)∩λ)∪μ. Then, x⊕fR⊕gR/fR=y⊕fR⊕gR/fR if and only if x⊕gR=y⊕gR, for all x,y∈R.Proof.
Assume thatx⊕fR⊕gR/fR=y⊕fR⊕gR/fR, for all x,y∈R. Then, we have that ((gR/fR((x-y)⊕fR))∩λ)∪μ=((gR/fR(0⊕fR))∩λ)∪μ=(gR(0)∩λ)∪μ, and so
(22)(∩{gR(t)∣t∈R,t⊕fR=(x-y)⊕fR}∩λ)∪μ=∩{(gR(t)∩λ)∪μ∣t∈R,t⊕fR=(x-y)⊕fR}=(gR(0)∩λ)∪μ.
Ift⊕fR=(x-y)⊕fR, then (fR(x-y-t)∩λ)∪μ=(fR(0)∩λ)∪μ. Therefore,
(23)(gR(x-y)∩λ)∪μ=(gR(x-y-t+t)∩λ)∪μ=(gR(x-y-t+t)∩λ∩λ)∪μ⊆((gR(x-y-t)∪gR(t)∪μ)∩λ)∪μ=((gR(x-y-t)∩λ)∪μ)∪((gR(t)∩λ)∪μ)⊆((fR(x-y-t)∩λ)∪μ)∪((gR(t)∩λ)∪μ)=((fR(0)∩λ)∪μ)∪((gR(t)∩λ)∪μ)⊆(gR(t)∩λ)∪μ.
It follows that (gR(x-y)∩λ)∪μ⊆∩{(gR(t)∩λ)∪μ∣t∈R,t⊕fR=(x-y)⊕fR} = (gR(0)∩λ)∪μ. On the other hand, we have (gR(0)∩λ)∪μ⊆(gR(x-y)∩λ)∪μ, by Lemma 11. Hence, (gR(x-y)∩λ)∪μ=(gR(0)∩λ)∪μ. And thus, x⊕gR=y⊕gR.
Conversely, suppose thatx⊕gR=y⊕gR, for all x,y∈R. It follows that (gR(x-y)∩λ)∪μ=(gR(0)∩λ)∪μ=(gR/fR(0⊕fR)∩λ)∪μ. Therefore,
(24)(gR((x-y)⊕fR)∩λ)∪μ=(∩{gR(t)∣t∈R,tt⊕fR=(x-y)⊕fR}∩λ)∪μ=∩{(gR(t)∩λ)∪μ∣t∈R,tt⊕fR=(x-y)⊕fR}=(gR(0)∩λ)∪μ=((gR/fR(0⊕fR))∩λ)∪μ;
that is, (gR/fR((x⊕fR)-(y⊕fR))∩λ)∪μ=(gR/fR(0⊕fR)∩λ)∪μ. It follows from Proposition 24 that x⊕fR⊕gR/fR=y⊕fR⊕gR/fR.Theorem 41 (third isomorphism theorem).
LetfR and gR be (λ,μ)-soft union ideals of R such that gR⊆~fR and (fR(0)∩λ)∪μ=(gR(0)∩λ)∪μ. If hR is a soft union ring, then (hR/fR)/(gR/fR)≅hR/gR.Proof.
LetΨ:(R/fR)/(R/fR)→R/gR be a mapping such that Ψ(r⊕fR⊕gR/fR)=r⊕gR, 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
(25)Ψ((x⊕fR⊕gR/fR)+(y⊕fR⊕gR/fR))=Ψ((x+y)⊕fR⊕gR/fR)=(x+y)⊕gR=(x⊕gR)+(y⊕gR),Ψ((x⊕fR⊕gR/fR)·(y⊕fR⊕gR/fR))=Ψ((xy)⊕fR⊕gR/fR)=(xy)⊕gR=(x⊕gR)·(y⊕gR).
Thus, Ψ is a ring isomorphism.
Moreover, since(26)((hR/fR)/(gR/fR))(x⊕fR⊕gR/fR)=∩{hR/fR(u⊕fR)∣u∈R,u⊕fR⊕gR/fR=x⊕fR⊕gR/fR}=∩{∩{hR(v)∣v∈R,v⊕fR=u⊕fR}∣u∈R,u⊕fR⊕gR/fR=x⊕fR⊕gR/fR}=∩{hR(v)∣v∈R,v⊕fR⊕gR/fR=x⊕fR⊕gR/fR}=∩{hR(v)∣v∈R,v⊕gR=x⊕gR}=(hR/gR)(x⊕gR)=(hR/gR)(Ψ(x⊕fR⊕gR/fR))=Ψ-1((hR/gR)(x⊕fR⊕gR/fR)),
that is, (hR/fR)/(gR/fR)=Ψ-1(hR/gR) therefore Ψ((hR/fR)/(gR/fR))=hR/gR, and so (hR/fR)/(gR/fR)≅hR/gR.


## 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.



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