Minimal prime ideals and cycles in annihilating-ideal graphs

Abstract

Let R be a commutative ring with identity, and let A(R) be the set of ideals with non-zero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R) * = A(R) \ {0}, and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study some connections between the graph theoretic properties of this graph and some algebraic properties of a commutative ring. We prove that if AG(R) is a tree, then either AG(R) is a star graph or a path of order 4 and, in the latter case, R = F S, where F is a field and S is a ring with a unique non-trivial ideal. Moreover, we prove that if R has at least three minimal prime ideals, then AG(R) is not a tree. It is shown that, for every reduced ring R, if R has at least three minimal prime ideals, then AG(R) contains a triangle. Also, we prove that, for every non-reduced ring R, if |Min (R)| = 2, then either AG(R) contains a cycle or AG(R) = P 4 . Finally, it is proved that, if |Min (R)| = 1 and AG(R) is a bipartite graph, then AG(R) is a star graph.