On the genus and crosscap two coannihilator graph of commutative rings

Abstract

Consider a commutative ring with unity denoted as $\mathscr {R}$, and let $W(\mathscr {R})$ represent the set of non-unit elements in $\mathscr {R}$. The coannihilator graph of $\mathscr {R}$, denoted as $AG'(\mathscr {R})$, is a graph defined on the vertex set $W(\mathscr {R})^*$. This graph captures the relationships among non-unit elements. Specifically, two distinct vertices, x and y, are connected in $AG'(\mathscr {R})$ if and only if either $x \notin xy\mathscr {R}$ or $y \notin xy\mathscr {R}$, where $w\mathscr {R}$ denotes the principal ideal generated by $w \in \mathscr {R}$. In the context of this paper, the primary objective is to systematically classify finite rings $\mathscr {R}$ based on distinct characteristics of their coannihilator graph. The focus is particularly on cases where the coannihilator graph exhibits a genus or crosscap of two. Additionally, the research endeavors to provide a comprehensive characterization of finite rings $\mathscr {R}$ for which the connihilator graph $AG'(\mathscr {R})$ attains an outerplanarity index of two.