Document Type: Original Manuscript

Author

Department of Mathematics, University of Yasouj, P.O.Box 75914, Yasouj, Iran.

Abstract

In this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. We
observe that over a commutative ring $R$, $Bbb{AG}_*(_RM)$ is
connected and diam$Bbb{AG}_*(_RM)leq 3$. Moreover, if $Bbb{AG}_*(_RM)$ contains a cycle, then $mbox{gr}Bbb{AG}_*(_RM)leq 4$. Also for an $R$-module $M$ with
$Bbb{A}_*(M)neq S(M)setminus {0}$, $Bbb{A}_*(M)=emptyset$
if and only if $M$ is a uniform module and ann$(M)$ is a prime
ideal of $R$.

Keywords