Isomorphisms and Planarity of Zero-Divisor Graphs

Abstract

Let R be a commutative ring with nonzero identity and I a proper ideal of R. The zero-divisor graph of R, denoted by \(\varGamma (R)\), is the graph on vertices \(R^*=R\setminus \{0\}\) where distinct vertices x and y are adjacent if and only if \(xy=0\). The ideal-based zero-divisor graph of R with respect to the ideal I, denoted by \(\varGamma _I(R)\), is the graph on vertices \(\{x \in R\setminus I \mid xy\in I\) for some \(y\in R\setminus I \}\), where distinct vertices x and y are adjacent if and only if \(xy\in I\). In this paper, we cover two main topics: isomorphisms and planarity of zero-divisor graphs. For each topic, we begin with a brief overview on past research on zero-divisor graphs. Whereafter, we provide extensions of that material to ideal-based zero-divisor graphs.