Decompositions And Zero-divisor Graphs Of Finite Commutative Group Rings

Group rings are important algebraic structures in ring theory.They are closely related to group theory,ring theory,field theory,algebraic topology and so on.In recent years,Group rings have been widely used in communication,cryptography and other fields.Let R be an associative ring with identity,G be a group and RG be the group ring.Maschke’s theorem states that RG is semisimple if and only if R is a semisimple ring,G is a finite group,and |G| is invertible in R.By Wedderburn-Artin theorem,we obtain the characterization of structure of semisimple group rings,that is,semisimple group rings are isomorphic to the direct product of finite matrix rings over division rings.It is difficult to study the structure of general group rings.In this paper,we study the decomposition and structure of a group ring RG where R is a finite commutative ring and G is a finite commutative group.In particular,when R is a finite commutative local ring and G is a cyclic group of prime power order,we give a description of the structure of RG.In recent years,the study on zero-divisor graphs of rings has become one of the hot topics in algebra research.The study of zero-divisor graph is of great significance to characterize the structure of algebraic systems.Its importance lies in that it can make the structure of algebraic systems more clear and intuitive.The properties of zero-divisor graphs of group rings ZnG have been described in literature,where Zn is a residue class ring modulo n and G is a finite commutative group.In this paper,we study the zero-divisor graphs of finite commutative group rings.This paper is composed of seven parts,where the first part is the introduction,the second to the sixth parts in which each part is a chapter are the core of the paper,and the last part is the summary.In Chapter 1,we mainly introduce the research background,theoretical sources and the development history of zero-divisor graph.In addition,we introduce some definitions and basic facts needed in this paper.In Chapter 2,we mainly study the decomposition and structure of finite commutative group rings.In particular,we give a description of the structure of group ring RG where R is a finite commutative local ring and G is a cyclic group of prime power order(Theorem 2.2.4).As an application,the structure of group rings ZpG is characterized(Corollary 2.2.7),where p is a prime and G is a cyclic group of prime power order.In Chapter 3,we mainly study the girth of zero-divisor graphs of group rings RG,where R is a finite commutative ring and G is a finite commutative group,and give characteriza-tion(Theorem 3.2.5).As an application,we can obtain the characterization of the girth of zero-divisor graph of a group ring Z,G(Corollary 3.2.6).In Chapter 4,we mainly study the planarity of zero-divisor graph of group rings RG where R is a finite commutative ring and G is a finite commutative group,and give char-acterization(Theorem 4.3.5).As an application,we can obtain the characterization of the planarity of zero-divisor graph of a group ring ZnG(Corollary 4.3.6).In Chapter 5,we mainly discuss the radius and diameter of zero-divisor graphs of group rings RG,where R is a finite commutative ring and G is a finite commutative group,and give characterizations(Theorem 5.2.3,Theorem 5.3.7).As an application,we can obtain the characterizations of the radius and diameter of zero-divisor graph of a group ring ZnG(Corollary 5.2.4,Corollary 5.3.8).In Chapter 6,we study the center and median of zero-divisor graph of a group ring ZnG.In particular,we describe the center and median of zero-divisor graph of a group ring ZpG,where p is a prime and G is a cyclic group of prime power order.In Chapter 7,we summarize the main results of this paper,and set out problems that could be further studied.