Rings whose annihilating-ideal graphs have positive genus (Q2892988)

The authors study genus \(\lambda(\mathbb{AG}(R) )\) of the annihilating graph \(\mathbb{AG}(R) \) of a commutative ring \(R\). For an artinian ring \(R\) with \(\lambda(\mathbb{AG}(R) )<\infty\), they prove that either \(R\) has a finite number of ideals, or \((R,\mathfrak{m})\) is Gorenstein with \(\dim_{R/\mathfrak{m}}\mathfrak{m}/\mathfrak{m}^2=2.\) For a non-domain noetherian local ring \(R\) with finite genus on \(\mathbb{AG}(R)\), it is proved that either \(R\) is Gorenstein or \(R\) is artinian with a finite number of ideals. For a fixed couple of integers \(m\geq 0, n>0\), it is proved that there exists only a finite number of isomorphic classes of artinian rings \(R\) such that \(\lambda(\mathbb{AG}(R))=m\) and \(|R/\mathfrak{m}|\leq n.\)