The classi cation of rings with its genus of class of graphs

Abstract

Let $R$ be a commutative ring, $I$ be a proper ideal of $R$, and $S(I)=\{a\in R : ra\in I \text{ for some } r\in R\sm I\}$ be the set of all elements of $R$ that are not prime to $I$. The total graph of $R$ with respect to $I$, denoted by $T(\Gamma_I(R))$, is the simple graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in S(I)$. In this paper, we determine all isomorphic classes of commutative Artinian rings whose ideal-based total graph has genus at most two.