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`The Scientific World JournalVolume 2014, Article ID 541630, 9 pageshttp://dx.doi.org/10.1155/2014/541630`
Research Article

## Soft Congruence Relations over Rings

Department of Mathematics, Northwest University, Xi’an 710127, China

Received 25 February 2014; Accepted 1 April 2014; Published 24 April 2014

Copyright © 2014 Xiaolong Xin and Wenting Li. 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

Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainty. In this paper, we initiate the study of soft congruence relations by using the soft set theory. The notions of soft quotient rings, generalized soft ideals and generalized soft quotient rings, are introduced, and several related properties are investigated. Also, we obtain a one-to-one correspondence between soft congruence relations and idealistic soft rings and a one-to-one correspondence between soft congruence relations and soft ideals. In particular, the first, second, and third soft isomorphism theorems are established, respectively.

#### 1. Introduction

To solve complicated problems in economics, engineering, environmental science, medical science, and social science, methods in classical mathematics are not always successfully used because various uncertainties are typical for these problems. Therefore, there has been a great deal of alternative research and applications in the literature concerning some special tools such as probability theory, fuzzy set theory [1, 2], rough set theory [3, 4], vague set theory [5], and interval mathematics [6]. However, all of these theories have their own difficulties which are pointed out in [7]. In 1999, Molodtsov [7] introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties.

Currently, works on soft set theory are progressing rapidly. Maji et al. [8] discussed the application of soft set theory to a decision making problem. Chen et al. [9] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. In theoretical aspects, Maji et al. [10] defined and studied several operations on soft sets, and Ali et al. [11] gave some new notions such as restricted intersection, restricted union, restricted difference, and extended intersection of soft sets. Sezgin and Atagün [12] discussed the basic properties of operations on soft sets such as intersection, extended intersection, restricted union, and restricted difference. Aktaş and Çaǧman [13] compared soft sets to the related concepts of fuzzy sets and rough sets. They also defined the notion of soft groups and derived some related properties. Furthermore, Jun [14] introduced and investigated the notion of soft BCK/BCI-algebras. We also noticed that Feng et al. [15] have already investigated the structure of soft semirings, and Acar et al. [16] have proposed the definition of soft rings and given some properties of soft rings. In [17, 18], Liu et al. have established three isomorphism theorems and fuzzy isomorphism theorems of soft rings. In fact, several researchers have investigated a fuzzy theory in soft structures (see [19]). At the same time, Majumdar and Samanta [20] introduced an idea of soft mapping. Moreover, Ali [21] generalized binary relations and proposed soft binary relations and soft equivalence relations. And Ali et al. [22] introduced algebraic structures of soft sets associated with new operations. In this paper, we extend soft binary relations over sets to soft congruence relations over rings.

The rest of the paper is organized as follows. In Section 2, some basic notions and results about soft sets are given. In Section 3, we introduce soft congruence relations and investigate several related properties. At the same time, a one-to-one correspondence between soft congruence relations and idealistic soft rings over rings is obtained. In Section 4, we obtain that the set of all soft congruence relations associated with some soft operations over a ring can form a complete lattice. And we consider the relations between soft congruence relations and homomorphisms over rings. Also, we establish the first soft isomorphism theorem. In Section 5, we obtain some related results of soft congruence relations of soft rings which is similar to soft congruence relations over rings and set up the second and third soft isomorphism theorems, respectively.

#### 2. Preliminaries

In this section, we recall some notions and definitions (see [7, 15, 22, 23]) that will be used in the sequel.

From now on, let be an initial universe and let be a set of parameters. Let denote the power set of and let and be nonempty subsets of .

Definition 1 (see [7]). A pair is called a soft set over , where is a mapping given by .

Definition 2 (see [7]). For two soft sets and over a common universe , we say that is a soft subset of if it satisfies(1);(2) for all .

We write .

In this case, is said to be a soft super set of .

Definition 3 (see [15]). The AND-operation of two soft sets and over a common universe is the soft set where and for all , . In this case, we write .

Definition 4 (see [15]). The OR-operation of two soft sets and over a common universe is the soft set where and for all , . In this case, we write .

Definition 5 (see [22]). The extended intersection of two soft sets and over a common universe is the soft set where and for all ,
In this case, we write .

Definition 6 (see [22]). The restricted intersection of two soft sets and over a common universe is the soft set where and for all , . In this case, we write .
If , then , where is the unique soft set over with an empty parameter set.

Definition 7 (see [22]). The extended union of two soft sets and over a common universe is the soft set where and for all ,
In this case, we write .

Definition 8 (see [22]). The restricted union of two soft sets and over a common universe is the soft set where and for all , . In this case, we write .
If , then .

In a similar way, we can define the AND-operation, the OR-operation, the extended intersection, the restricted intersection, the extended union, and the restricted union of a family of soft sets over as follows in [23].

#### 3. Soft Congruence Relations over Rings

In this section, we will introduce the notion of soft congruence relations and investigate several related properties. From now on, denotes a ring.

Let be a soft set over . The set is called the support of the soft set . A soft set is said to be nonnull if (see [15]).

Let us recall some definitions about soft relations (see [21]) that we will use in the following paragraphs.

Definition 9. Let be a soft set over . Then is called a soft binary relation over .

Definition 10. A soft binary relation over is called a soft equivalence relation over if is an equivalence relation on for all .

Definition 11. An equivalence relation over is called a congruence relation on if and can imply and for all .

Let us define now a soft congruence relation over .

Definition 12. A nonnull soft set over is called a soft congruence relation over if is a congruence relation on for all .
If , then is said to be a null soft congruence relation over , denoted by .

Let be a soft congruence relation over . If the parameter set is a singleton set, is equivalent to a classical congruence relation. If not, is a general soft congruence relation. That is, the classical congruence can be considered as a soft congruence relation. Then we can give the following two examples of general soft congruence relations.

Example 13. Let be a ring and . Let us consider the set-valued function given by for all ; that is, is a congruence on module . It is clear that is a congruence relation on . Hence is a soft congruence relation over .

Example 14. Let and let be a set-valued function defined by for all , where is the principal ideal generated by . It is easy to verify that is a congruence relation on . Hence is a soft congruence relation over .

Theorem 15. Let and be soft congruence relations over . Then(1) is a soft congruence relation over if it is nonnull;(2) is a soft congruence relation over ;(3) is a soft congruence relation over if it is nonnull.

Proof. (1) By Definition 3, let , where and for all . Then by the hypothesis, is a nonnull soft set over . If , then . It follows that the nonempty sets and are both congruence relations on . Hence is a congruence relation on for all and so is a soft congruence relation over .
(2) By Definition 5, let , where , and for all ,
Let . If , then is a congruence relation on ; if , then is a congruence relation on ; and if , . Thus and are both congruence relations on and so is their intersection. Hence is a congruence relation on for all . It follows that is a soft congruence relation over .
(3) The proof is similar to (1).

Theorem 16. Let and be soft congruence relations over . Then(1) is a soft congruence relation over if or for all ;(2) is a soft congruence relation over if or for all ;(3) is a soft congruence relation over if or for all with .

Proof. (1) By Definition 4, let = , where and = for all . Let ; then = . Then by the hypothesis, we know that or for all . It follows that the nonempty sets and are both congruence relations on . Hence is a congruence relation on for all and so = is a soft congruence relation over .
(2) By Definition 7, let = , where , and for all ,
Let Supp. If , then is a congruence relation on ; if , then is a congruence relation on ; and if , = . Then by the hypothesis, we know that or for all . Hence is a congruence relation on for all Supp. It follows that = is a soft congruence relation over .
(3) The proof is similar to (1).

In a similar way, we can obtain similar properties associated with the AND-operation, the extended intersection, the restricted intersection, the OR-operation, the extended union, and the restricted union of a family of soft congruence relations over .

Then, we will consider relations between soft congruence relations and idealistic soft rings over rings. In order to do this, we recall the following notions and results.

If there exists an element such that and for all . Then is called the zero of . From now on, let be the zero of .

Definition 17 (see [16]). Let be a nonnull soft set over . Then is called a soft ring over , denoted by , if is a subring of for all .

Definition 18 (see [16]). Let be a nonnull soft set over . Then is called an idealistic soft ring over if is an ideal of for all .

Lemma 19 (see [24]). Let be a congruence relation on . Then is an ideal of and . Conversely, let be an ideal of and define in . Then is a congruence relation and .

Definition 20. Let be a soft congruence relation over . For , we can consider the set-valued function given by for all , where is the congruence class of with respect to . We say that is a soft congruence class of with respect to .

Definition 21. Let be a soft congruence relation over and let be an initial universe set. Let us consider the set-valued function given by for all . We say that is a soft quotient set of . For all , is a quotient ring of with respect to ; we say that is a soft quotient ring of with respect to the soft congruence relation . Here, for , we have and for all .

Example 22. Let be a ring with the operation tables given in (5). For , let be a set-valued function defined by for all . Then and , which are both congruence relations on . Hence is a soft congruence relation over . And let be a soft quotient ring of with respect to .
The operation equations of the ring are as follows: For , we have soft congruence classes: Soft quotient ring is as follows:

The next two theorems show connections between soft congruence relations and idealistic soft rings over rings.

Theorem 23. (1) Let be a soft congruence relation over . If , then is an idealistic soft ring over , and we have for all .
(2) Let be an idealistic soft ring over and consider the set-valued function defined by for all . Then is a soft congruence relation over and .

Proof. (1) By Definition 12, we know that, for all , is a congruence relation on . And according to Lemma 19, we have that is an ideal of , and define in . Since is a nonnull soft set, we have that is an idealistic soft ring over by Definition 17 and for all .
(2) By Definition 18, we know that, for all , is an ideal of . And according to Lemma 19, we have that is a congruence relation on and for all . Hence, is a soft congruence relation over and .
Here, we obtain that any soft congruence relation over can be represented by the idealistic soft ring generated by . Also, we observe that any idealistic soft ring over is the soft congruence class of with respect to the soft congruence relation generated by .

Summarizing Theorem 23, we have the following facts.

Denote by the set of all soft congruence relations and by the set of all idealistic soft rings over . We can establish the following two mappings:(1), where ;(2), where for all , we define in .

Theorem 24. The above two mappings and are inverse mappings. Then there is a one-to-one correspondence between and over .

#### 4. Soft Congruence Relations and Homomorphisms

At the beginning of this section, we study the set of all soft congruence relations associated with some soft operations over a ring which can form a complete lattice.

Definition 25. A soft congruence relation over is said to be trivial, denoted by , if for all . A soft congruence relation over is said to be whole, denoted by , if for all .

Example 26. Let and be nonnull soft sets over . For all , we have . For , it is easy to verify that is a soft congruence over , and for all , . Then is the trivial soft congruence over . For , it is clear that is a soft congruence over and . Then is the whole soft congruence over .
In a similar way, we can define the whole soft congruence relation with respect to the set of parameters which is called the absolute soft congruence over and simply denoted by . And, we will denote by the unique soft congruence over with an empty parameter set, which is called the empty soft congruence over .

Now we consider whether the set of all soft congruence relations associated with some soft operations over a ring can form a complete lattice. For the concepts and results of lattices, we can see reference [25].

Let be a nonnull soft set over . We call the smallest soft congruence relation which is the soft super set of to be a soft congruence relation generated by , denoted by , namely, the restricted intersection of the family of all soft congruence relations over which are soft super sets of . In this case, we write .

Let and . We denote . Then is a complete lattice, where and are the greatest element and the least element of , respectively.

In a similar way, denote by the set of all those soft congruence relations defined over with a fixed parameter set . Then is a complete lattice, where and are the greatest element and the least element of , respectively.

Next, we will consider the relations between soft congruence relations and homomorphisms over rings.

Lemma 27. Let be a ring epimorphism and let be a congruence relation on . Define ; then is a congruence relation on .

Proof. For all , and , since is a ring epimorphism, there exists , such that , and . It is easy to verify that is a congruence relation on .

Let be a mapping of rings and let be a soft set over . Then we can define a soft set over where is defined as for all . Here, by definition, we see that = .

Lemma 28. Let be a ring epimorphism. If is a soft congruence relation over , then is a soft congruence relation over , where for all .

Proof. Note first that is a nonnull soft set since is a soft congruence relation over , which is a nonnull soft set by Definition 11. For all , we have . Then the nonempty set is a congruence relation on , and so we deduce that its onto homomorphic image is a congruence on . Hence is a congruence on for all . That is, is a soft congruence relation over .

Lemma 29. Let be a ring epimorphism and let be a soft congruence relation over , where for all .(1)If is trivial, then is the trivial soft congruence relation over .(2)If is whole, then is the whole soft congruence relation over .

Proof. (1) By Definition 25, we know that, for all , . Since is a ring epimorphism, we have that for all . So, is the trivial soft congruence relation over by Lemma 28.
(2) By Definition 25, we know that, for all , . Since is a ring epimorphism, we have that for all . It follows from Lemma 28 that is the whole soft congruence relation over .
This completes the proof.

Let us define now some definitions about soft quotient rings that we will use in the following paragraphs.

Definition 30. Let be a soft congruence relation over ; we have that is an idealistic soft ring over by Theorem 23, and for all , . Let be an initial universe set and consider the set-valued function given by for all . We say that is a soft quotient ring of with respect to the idealistic soft ring , where is a quotient ring of with respect to an ideal .

Example 31. Let be an idealistic soft ring over and . For all , we have . Let be an initial universe set and consider the set-valued function given by for all ; it is clear that is a quotient ring of with respect to . Then is a soft quotient ring of with respect to .

Definition 32. Let be a soft quotient ring of with respect to an idealistic soft ring and let be a soft quotient ring of with respect to an idealistic soft ring , respectively. We say that is soft homomorphic to , denoted by , if, for all , there exists such that a mapping is a homomorphism; that is, .

If is an isomorphism, then we say that is soft isomorphic to , which is denoted by .

Finally, we establish the first soft isomorphism theorem. In order to do this, we need to introduce some relative notions and results. Note that, if is a subring of , we write ; if is an ideal of , we write .

Definition 33. Let and be soft sets over a common universe . is said to be contained by , denoted by if, for all , there exists such that .

Definition 34. Two soft sets and over a common universe are said to be soft set equal, denoted by , if it satisfies the following:(1)for all , such that ;(2)for all , such that .

Example 35. Let , , and . Let and be set-valued functions defined as follows: , , , and . Therefore, and are soft set equal.

Lemma 36. Let be a ring epimorphism.(1)Let be a set-valued function given by for all . Then is a soft congruence relation over .(2)Let be a set-valued function given by for all . Then is an idealistic soft ring over .

Lemma 37 (see [24] first isomorphism theorem). Let be a ring epimorphism. Let and . Then, for , we make correspond to , which is a bijective mapping between and . And ; here, .

Theorem 38 (first soft isomorphism theorem). Let be a ring epimorphism and consider an idealistic soft ring over defined as Lemma 36(2). Let , , and .(1)For , let be a set-valued function given by for all . We make correspond to , which is a bijective mapping between and .(2) is an idealistic soft ring over if and only if is an idealistic soft ring over . And , where is a soft quotient ring of with respect to , and is a soft quotient ring of with respect to .

Proof. (1) Let if ; that is, for all , such that and for all , such that , but by Lemma 37, we have and , which means that . Thus it is injection. Let ; then we have, for all , and its homomorphism image , so ; that is, . Let ; then we have for all , . Define for all ; then . And for all , , Ker . So and ; that is, . Thus it is surjection. Therefore, we make correspond to , which is a bijective mapping between and .
(2) is an idealistic soft ring over (for all (by Lemma 37)   for all is an idealistic soft ring over . From Lemma 37, for all , which means that by Definition 32.

Example 39. Let be a ring epimorphism, and . Consider the set-valued function given by . Then is an idealistic soft ring over . Let , . Then for , we have . And let . Then for , we have .(1)Let and let be a set-valued function given by for all . Then, . It is clear that we make correspond to , which is a bijective mapping between and .(2)Let and . Then is an idealistic soft ring over iff (for all , (for all ), and is an idealistic soft ring over . It is clear that is a soft quotient ring of with respect to , and is a soft quotient ring of with respect to . Let ; we have and . Note that and ; we have by Definition 32.

#### 5. Soft Congruence Relation of Soft Rings

In this section, we study internal connections between soft congruence relations and soft ideals of soft rings and obtain the second and third soft isomorphism theorems. In order to do this, we recall the following notions.

Definition 40 (see [16]). Let and be soft rings over . Then is called a soft subring of , denoted by , if it satisfies the following:(1);(2) is a subring of for all .

Definition 41 (see [16]). Let be a soft ring over . A nonnull soft set over is called a soft ideal of , denoted by , if it satisfies the following:(1);(2) is an ideal of for all .

Definition 42. Let be a soft ring over . A nonnull soft set over is called a soft binary relation of if it satisfies the following:(1);(2) is a binary relation of for all .

Definition 43. Let be a soft ring over . A soft binary relation of is called a soft congruence relation of if is a congruence relation on for all .

The next theorem shows connections between soft congruence relations and soft ideals of soft rings.

Theorem 44. Let be a soft ring over . Then we have the following.(1)Let be a soft congruence relation of . If , then is a soft ideal of , and we have for all .(2)Let be a soft ideal of and consider a soft binary relation of defined by for all . Then is a soft congruence relation of , and .

Proof. (1) By Definition 43, we know that, for all , is a congruence relation on a ring . Since , we have , and for all by Lemma 19. Since is a non-null soft set, we have .
(2) By Definition 41, we know for all , . By the hypothesis, we know that is a congruence relation on and by Lemma 19. Then is a soft congruence relation of and .
Here, we obtain that any soft congruence relation of a soft ring can be represented by the soft ideal generated by . Also, we observe that any soft ideal of is the soft congruence class of with respect to the soft congruence relation generated by .

Summarizing Theorem 44, we have the following facts.

Denote by the set of all soft congruence relations and by the set of all soft ideals of a soft ring . We can establish the following two mappings:(1), , where ;(2), , where, for all , we define in , which is equivalent to .

Theorem 45. The above two mappings and are inverse mappings. Then there is a one-to-one correspondence between and of .

Definition 46. Let and be soft rings over . We say that is a generalized soft subring of , denoted by , if, for all , there exists such that is a subring of .

Definition 47. Let and be soft rings over . We say that is a generalized soft ideal of , denoted by , if, for all , there exists such that is an ideal of .

Here, we obtain that a soft ideal of soft ring must be a generalized soft ideal of soft ring.

Definition 48. Let be a generalized soft ideal of a soft ring which is over and let be an initial universe set, where . Let us consider the set-valued function given by for all . We say that is a generalized soft quotient ring of soft ring with respect to the generalized soft ideal , where is a quotient ring of ring with respect to .

Example 49. Let and . Let us consider the set-valued function given by . Then , , , and . As we see, all these sets are subrings of . Hence, is a soft ring over . On the other hand, consider the function given by . As we see,
Hence, is a generalized soft ideal of . For all , where , ; we have
As we see, are quotient rings of ring with respect to . Thus is a generalized soft quotient ring of soft ring with respect to the generalized soft ideal .

Definition 50. Let and be generalized soft ideals of soft rings and , respectively. Suppose is a generalized soft quotient ring of with respect to , and is a generalized soft quotient ring of with respect to . We say that is soft homomorphic to , denoted by , if, for all , there exists such that a mapping is a homomorphism; that is, .

If is an isomorphism, then we say that is soft isomorphic to , which is denoted by .

Definition 51 (see[16]). Let and be soft rings over the rings and , respectively. Let and be two mappings. The pair is called a soft ring homomorphism if the following conditions are satisfied:(1) is a ring epimorphism;(2) is surjective;(3) for all .

If we have a soft ring homomorphism between and , is said to be soft homomorphic to , denoted by . In addition, if is a ring isomorphism and is a bijective mapping, then is called a soft ring isomorphism. In this case, we say that is soft isomorphic to , denoted by .

Definition 52. The basic sum of two soft rings and over is the soft set where and for all , . In this case, we write = .

Lemma 53. Let and be soft rings over rings and , respectively. Let be a soft ring homomorphism between and , and .(1)Let us consider a soft binary relation of defined by for all . Then is a soft congruence relation of .(2)Let be a set-valued function given by for all . Then is a soft ideal of .

Theorem 54 (second soft isomorphism theorem). Let and be soft rings over rings and , respectively. Let be a soft ring epimorphism between and , and consider a soft ideal of defined as Lemma 53(2). Denote and .(1)Let and consider the set-valued function given by for all . Then we make correspond to , which is a bijective mapping between and .(2) is a soft ideal of if and only if is a generalized soft ideal of . And , where is a generalized soft quotient ring of with respect to , and is a generalized soft quotient ring of with respect to .

Proof. (1) Let . If , that is, for all , such that and for all , such that , but by Lemma 37, we have and , which means that . Thus it is injection. Let ; we have that, for all , ; then its homomorphism image , and so ; that is, . Let , we have that, for all , such that . So we can define ; then for all , and for all , . So and