The Graphic Structure And Algebraic Structure Of Rings And Semigroups

Zero-divisor graph is a new area of research developed in recent years,which mainly studies the relationship between the algebraic structures of rings or semigroups and the graphic structures and properties of their zero-divisor graphs.In this paper,we study the algebraic structures and properties of commutative rings and semigroups by taking advantage of their zero-divisor graphs,and we are mainly interested in the structure of non-isomorphic commutative semigroups and rings whose zero-divisor graph is a refinement of a star graph or a graph which contains no rectangles.For a refinement G of a star graph with center c,let G_c^* be the subgraph of G induced on the vertex set V(G)\{c and end vertices adjacent to c}.This paper consists of the following parts.In Chapter 0,we introduce some backgrounds and known results,and give a brief introduction to the main results of our work.In Chapter 1,we study the algebraic properties of nilpotent semigroups whose zero-divisor graph is a refinement of a star graph,and answer a question which we have posed in ref.[1]that which class B of graphs has the property that each G∈B has a unique S such thatΓ(S)≌C and Sⁿ={0},S^n-1≠{0}? For each finite n≥5,we construct a nilpotent semigroup S such thatΓ(S) is a refinement of a star graph with exactly one center andΓ(S)\T is a refinement of K_1,n-3,where Sⁿ={0},S^n-1≠{0} and T consists of end vertices adjacent to the center.In Chapter 2 and Chapter 3,we study the isomorphic classification problem of finite local rings by investigating their zero-divisor graphs,and this method is different from the past.D.F.Anderson and P.S.Livingston have proved in ref.[2] that the zero-divisor graph of any finite local ring is a refinement of a star graph, and the corresponding ring of every finite proper refinement of star graphs is local. We make our investigations based on this result.For a proper refinementΓ(R) of a star graph with center c,there are only two possible cases,that is,eitherΓ(R)_c^* has at least two connected components orΓ(R)_c^* is connected.We prove in Chapter 2 that ifΓ(R)_c^* has at least two connected components,then the ideal Z(R) is generated by two elementsα₁,α₂,whereα₁,α₂ are in distinct components ofΓ(R)_c^*(Theorem 2.3.3).Based on this result,it is proved that R must be finite and the algebraic structures of all such rings R are obtained.In order to consider the case thatΓ(R)_c^* is connected,we introduce a new concept c-local rings,and prove that the diameter of the induced graphΓ(R)_c^* is two ifΓ(R)_c^* is connected and R is c-local(Theorem 2.3.9).Furthermore,in Chapter 3,it is proved that for all finite c-local rings R,the maximal ideal Z(R) has a minimal generating set which has a c-partition(Theorem 3.2.6).In virtue of this result,the structure and classification up to isomorphism of all finite c-local rings are determined.Chapter 4 answers a question of Lu Dancheng and Wu Tongsuo:how can one characterize the zero-divisor graphs which contain no rectangles? We prove that a graph which contains no rectangles is a zero-divisor graph of a semigroup with 0 if and only if it is one of the following graphs:an isolated vertex,a star graph, a two-star graph,a triangle with n horns(n=0,1,2,3),a fan graph,a fan graph with a horn adjacent to its center(Theorem 4.2.2).In addition,we completely determine the correspondence between commutative rings and zero-divisor graphs which contain no rectangles(except the infinite star graphs).