The undirected annihilating-ideal graphs over non-commutative rings

For a ring R (not necessarily commutative) with identity, let \(\mathcal {A}(R)\) $\mathcal{A}(R)$ be the set of all annihilating-ideals of R. We define an undirected annihilating-ideal graph \(\mathcal{A}\mathcal{G}(R)\) $\mathcal{A}\mathcal{G}(R)$ of R with the vertex set \(\mathcal {A}(R)^{*}=\mathcal {A}(R)\backslash \{(0)\}\) $\mathcal{A}(R)_{}^{\ast }=\mathcal{A}(R)\backslash \{(0)\}$ , and two distinct vertices I and J are adjacent if and only if either \(IJ=(0)\) $IJ=(0)$ or \(JI=(0)\) $JI=(0)$ . In this paper, we investigate the connectedness, diameter and girth of this graph. We prove that if R is a duo ring such that \(\mathcal {A}(R)^{*}\ne \emptyset \) $\mathcal{A}(R)_{}^{\ast }\ne \mathrm{\varnothing }$ , then \(\mathcal{A}\mathcal{G}(R)\) $\mathcal{A}\mathcal{G}(R)$ has n vertices if and only if R has only n nonzero proper ideals. In addition, we also characterize those rings R for which \(\mathcal{A}\mathcal{G}(R)\) $\mathcal{A}\mathcal{G}(R)$ is a star graph, complete graph or complete bipartite graph.