# Classification of rings with toroidal annihilating-ideal graph

Document Type : Original paper

Authors

1 Department of Mathematics Manonmaniam Sundaranar University Tirunelveli

2 Manonmaniam Sundaranar University

Abstract

Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be the set of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, an annihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ =(0)$. The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertex set $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$ for which $AG(R)$ has genus one.

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