On the genus of graphs from commutative rings

Abstract

Let be a commutative ring with non-zero identity. The cozero-divisor graph of , denoted by , is a graph with vertex-set , which is the set of all non-zero non-unit elements of , and two distinct vertices and in are adjacent if and only if and , where for , is the ideal generated by . In this paper, we determine all isomorphism classes of finite commutative rings with identity whose has genus one. Also we characterize all non-local rings for which the reduced cozero-divisor graph is planar.

Keywords

• genus
• local ring
• nilpotent
• planar graph