Applications of Soft Union Sets in the Ring Theory

Journal of Applied Mathematics, Dec 2013

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.

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Applications of Soft Union Sets in the Ring Theory

Applications of Soft Union Sets in the Ring Theory Yongwei Yang, Xiaolong Xin, and Pengfei He Department of Mathematics, Northwest University, Xi'an 710127, China Received 3 July 2013; Accepted 28 October 2013 Academic Editor: Zhihong Guan Copyright © 2013 Yongwei Yang et al. 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 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 sati (...truncated)


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Yongwei Yang, Xiaolong Xin, Pengfei He. Applications of Soft Union Sets in the Ring Theory, Journal of Applied Mathematics, 2013, 2013, DOI: 10.1155/2013/474890