On the girth of the annihilating-ideal graph of a commutative ring

Subject Areas : History and biography

M. Ahrari ¹ , Sh. A. Safari Sabet ² , B. Amini ³

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
3 - Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran

Received: 2015-04-05 Accepted : 2015-12-13 Published : 2015-08-01

Keywords: annihilating-ideal graph, star graph, bipartite graph, girth,

Abstract :

The annihilating-ideal graph of a commutative ring $R$ isdenoted by $AG(R)$, whose vertices are all nonzero ideals of $R$with nonzero annihilators and two distinct vertices $I$ and $J$are adjacent if and only if $IJ=0$. In this article, we completelycharacterize rings $R$ when $gr(AG(R))\neq 3$.

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