COMMUTATIVE RINGS R WHOSE C(AG(R)) CONSIST OF ONLY TRIANGLES

摘要

Let R be a commutative ring with identity. The set I(R) of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of I(R) is called the annihilating-ideal graph of R, denoted by AG-(R). We write G for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in G that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that AG-(R) is an element of G is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with AG-(R) is an element of G. Also, a connection between finite rings and their corresponding graphs is realized.