Zero-divisor Graphs Of Group Rings Of The Dihedral Group

Group rings are very important algebraic structures in ring theory. Besides the obvious relationship with group theory and ring theory, they are related to field theory, linear algebra, algebraic number theory, algebraic topology etc. Recently, group rings have been widely applied to the fields of codes and communications. The study of algebraic structures, using the properties of graphs, becomes an exciting research topic in the last twenty years, leading to many fascinating results and questions. This paper mainly discusses algebraic properties, structures, and properties of zero-divisor graphs with regard to the noncommutative group rings ZnDm, where Zn is the residue class ring modulo n and Dm is a dihedral group of order 2m.This paper is composed of six parts, where the first part is the introduction, the second to the fifth parts in which each part is a chapter are the core of the paper, and the last part is the concluding remarks.In Chapter 1 of this paper, we summarize the history of the zero-divisor graph, the research background, theory source and research significance of this paper. At the same time, we give the notation and basic results of ring theory and graph theory.In Chapter 2, we investigate the algebraic properties of noncommutative group rings ZnDm and their structures. Specially, when m=p and m=2t, we give the detailed characterization to their algebraic structures [Theorem 2.2.1, Proposition 2.2.1, Corollary 2.2.1, Theorem 2.3.1, Proposition 2.3.1, Corollary 2.3.1].In Chapter 3, we mainly discuss the girth, the diameter and the planarity of zero-divisor graph of ZnDm[Theorem 3.1.1, Theorem 3.2.1, Theorem 3.2.2, Theorem 3.2.3]. Specially, we give the center of the undirected zero-divisors graph, when m—2t+1,(t is a positive integer)[Theorem 3.3.1].In Chapter 4, we mainly investigate the algebraic structures of group rings ZnD4 and their zero-divisors. At the same time, we give the detailed characterizations to them re-spectively[Theorem4.1.1, Theorem 4.1.2, Corollary 4.1.2, Corollary 4.1.3], where D4 is a dihedral group of order 8. Also, we give the properties of zero-divisor graph of group rings ZnD4. The main results of this section have been accepted and will be published in Journal of Guangxi Normal University.In Chapter 5, we mainly discuss the algebraic structures and zero-divisors of group ring ZnG, when ZnG is a finite local group ring[Theorem5.1.1, Theorem 5.1.2]. At the same time we discuss the girth, the diameter, the planarity and center of zero-divisor graph of ZnG[Theorem 5.2.1, Theorem 5.2.2, Theorem 5.2.3, Theorem 5.2.4]. The main results of this section have been published in Journal of Guangxi Normal University for Nationalities.The last part is the concluding remarks, the main work of this paper is summarized, some subjects corresponding to this paper are simply illustrated, and the idea to the next phase of work to continue to study is roughly envisaged.