Some Results on Annihilating-ideal Graphs

Abstract

The annihilating-ideal graph of a commutative ring $R$, denoted by $\mathbb{A}\mathbb{G}\left( R \right)$, is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\,=\,\left( 0 \right)$. Here we show that if $R$ is a reduced ring and the independence number of $\mathbb{A}\mathbb{G}\left( R \right)$ is finite, then the edge chromatic number of $\mathbb{A}\mathbb{G}\left( R \right)$ equals its maximum degree and this number equals ${{2}^{\left| \min \left( R \right) \right|-1}}-\,1$; also, it is proved that the independence number of $\mathbb{A}\mathbb{G}\left( R \right)$ equals ${{2}^{\left| \min \left( R \right) \right|-1}}$, where $\min \left( R \right)$ denotes the set of minimal prime ideals of $R$. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a finite graph $\mathbb{A}\mathbb{G}\left( R \right)$ is not Eulerian, and that it is Hamiltonian if and only if $R$ contains no Gorenstain ring as its direct summand.