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.