On the Genus and Crosscap of the Extended Sum Annihilating-Ideal Graph of Commutative Rings

Abstract

In the context of a commutative ring with unity, denoted as $\mathcal {S}$, and its associated set of annihilating ideals $A(\mathcal {S})$, there exists a graph known as the extended sum annihilating-ideal graph, denoted as $AG_\varOmega (\mathcal {S})$. This graph has its vertex set derived from the set $A(\mathcal {S})^*$, and it exhibits a specific pattern of connections between its vertices. More precisely, two distinct vertices, referred to as $\Im _1$ and $\Im _2$, are linked by an edge if and only if one of the following conditions holds: either $\Im _1\Im _2 = 0$ or $\Im _1 + \Im _2 \in A(\mathcal {S})$. In the following research paper, we delve into the classification of Artinian commutative rings, denoted as $\mathcal {S}$, with a particular focus on those where the extended sum annihilating-ideal graph takes on one of three distinct forms: a double toroidal graph, a projective plane, or a Klein bottle.