Commutative Rings And Their Annihilating-ideal Graphs

The annihilating-ideal graphs were first introduced and studied by M. Behboodirecently. The graphs provide an excellent setting for studying some aspects of alge-braic properties of commutative rings, especially, the ideal structures of rings. Themain topics of this dissertation are the realization of the zero-divisor graph of boundedsemirings, the relationship between the algebraic properties and structures of commu-tative rings and the graphic properties and structures of their annihilating-ideal graphs,and the realization of the connections between the rings and the corresponding graphs.This paper consists of the following parts.Some basic definitions and properties for commutative rings and graphs are in-troduced in Chapter1. we also fix some notations which are used frequently later.In Chapter2, we mainly study the zero-divisor graphs of bounded semirings.Based on some results about the zero-divisor graphs of semigroups, we investigate thetypes of bipartite graphs, complete graph with horns and the graphs whose cores con-sist only of triangles which can be realized as annihilating-ideal graphs of commutativerings.After a thorough investigation of zero-divisor graph of bounded semirings, westudy the annihilating-ideal graphs of rings. Chapter3is devoted to classifying thestructures of finite principal ideal rings in terms of the quotient of polynomial ringsover finite fields. As applications, we determine the correspondence between graphscontaining at most3vertices and finite rings. For the bipartite graphs and completegraph with horns, the realizations are deduced in Chapter4. In particular, it is provedthat a triangle with two horns cannot be realizable. Then the types of the graphs whosecores only consist of triangles which can be realized as annihilating-ideal graphs ofrings are determined in chapter5. The structure of the corresponding finite rings are studied in a way similar to that used in Chapter4. Finally, we grasp the common prop-erties T1and T2of local but not principal ideal rings studied in the last two chapters,and show that the rings satisfying conditions T1and T2are exactly the non-principalideal rings whose annihilating-ideal graphs are either star graph or these with coresconsist only of triangles.