On the genus of strong annihilating-ideal graph of commutative rings (Q6650692)

Let \(R\) be a commutative ring with unity and \(A(R)^*\) be the set of all nonzero annihilating-ideals of \(R.\) An ideal \(I\) of \(R\) is said to be annihilator ideal if there is a nonzero ideal \(J\) of \(R\) such that \(IJ = (0).\) For an ideal \(I\) of \(R\), \(\mathrm{Ann}(I)=\{r\in R: rI=(0)\}.\) The strong annihilating-ideal graph of \(R,\) denoted by \(\mathrm{SAG}(R),\) is an undirected simple graph with vertex set \(A(R)^*\) and two vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1\cap \mathrm{Ann}(I_2)\neq (0)\) and \(I_2\cap \mathrm{Ann}(I_1)\neq (0).\) In this paper, first authors characterize commutative Artinian rings whose strong annihilating-ideal graph is isomorphic to some well-known graphs and then they classify all commutative Artinian rings whose strong annihilating-ideal graph is planar, toroidal or projective.