Extended zero-divisor graphs of idealizations
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Let $R$ be a commutative ring with zero-divisors $Z(R)$. The extended zero-divisor graph of $R$, denoted by $\overline{\Gamma}(R)$, is the (simple) graph with vertices $Z(R)^*=Z(R)\backslash\{0\}$, the set of nonzero zero-divisors of $R$, where two distinct nonzero zero-divisors $x$ and $y$ are adjacent whenever there exist two non-negative integers $n$ and $m$ such that $x^ny^m=0$ with $x^n\neq 0$ and $y^m\neq 0$. In this paper, we consider the extended zero-divisor graphs of idealizations $R\ltimes M$ (where $M$ is an $R$-module). At first, we distinguish when $\overline{\Gamma}(R\ltimes M)$ and the classical zero-divisor graph $\Gamma(R\ltimes M)$ coincide. Various examples in this context are given. Among other things, the diameter and the girth of $\overline{\Gamma}(R\ltimes M)$ are also studied.
Abstract
Let $R$ be a commutative ring with zero-divisors $Z(R)$. The extended zero-divisor graph of $R$, denoted by $\overline{\Gamma}(R)$, is the (simple) graph with vertices $Z(R)^*=Z(R)\backslash\{0\}$, the set of nonzero zero-divisors of $R$, where two distinct nonzero zero-divisors $x$ and $y$ are adjacent whenever there exist two non-negative integers $n$ and $m$ such that $x^ny^m=0$ with $x^n\neq 0$ and $y^m\neq 0$. In this paper, we consider the extended zero-divisor graphs of idealizations $R\ltimes M$ (where $M$ is an $R$-module). At first, we distinguish when $\overline{\Gamma}(R\ltimes M)$ and the classical zero-divisor graph $\Gamma(R\ltimes M)$ coincide. Various examples in this context are given. Among other things, the diameter and the girth of $\overline{\Gamma}(R\ltimes M)$ are also studied.
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