The Annihilating-Ideal Graph of Commutative Rings II

Abstract

In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in Part I (see [5]). Let $R$ be a commutative ring with $\mathbb{A}(R)$ its set of ideals with nonzero annihilator and $Z(R)$ its set of zero divisors. The annihilating-ideal graph of $R$ is defined as the (undirected) graph $\mathbb{AG}(R)$ that its vertices are $\mathbb{A}(R{)}^{\ast }=\mathbb{A}(R)\setminus \{(0)\}$ in which for every distinct vertices $I$ and $J$, $I---J$ is an edge if and only if $IJ=(0)$. First, we study the diameter of $\mathbb{AG}(R)$. A complete characterization for the possible diameter is given exclusively in terms of the ideals of $R$ when either $R$ is a Noetherian ring or $Z(R)$ is not an ideal of $R$. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either $\chi (\mathbb{AG}(R))\le 2$ or $R$ is reduced and $\chi (\mathbb{AG}(R))\le \infty$. Also it is shown that for each reduced ring $R$, $\chi (\mathbb{AG}(R))=cl(\mathbb{AG}(R))$. Moreover, if $\chi (\mathbb{AG}(R))$ is finite, then $R$ has a finite number of minimal primes, and if $n$ is this number, then $\chi (\mathbb{AG}(R))=cl(\mathbb{AG}(R))=n$. Finally, we show that for a Noetherian ring $R$, $cl(\mathbb{AG}(R))$ is finite if and only if for every ideal $I$ of $R$ with $I^2=(0)$, $I$ has finite number of $R$-submodules.

arXiv comment16 peges, to appear in Journal of Algebra and its Applications