More on the annihilator graph of a commutative ring
Let \(R\) be a commutative ring with identity and \(Z(R)^*\) be the set of nonzero zero-divisors of \(R\). The zero-divisor graph of \(R\), denoted by \(\Gamma(R)\), is an undirected graph with vertex set \(Z(R)^*\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy = 0\). The annihilator graph of \(R\) is defined as an undirected graph \(AG(R)\) with vertex set \(Z(R)^*\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(\mathrm{ann}_R(xy) \neq \mathrm{ann}_R(x) \cup \mathrm{ann}_R(y)\). Clearly, if \(xy = 0\) for \(x, y \in Z(R)^*\), then \(\mathrm{ann}_R(x) \cup \mathrm{ann}_R(y) \subsetneq R = \mathrm{ann}_R(xy)\). Hence, \(\Gamma(R)\) is a subgraph of \(AG(R)\). In [Commun. Algebra 42, No. 1, 108--121 (2014; Zbl 1295.13006)], \textit{A. Badawi} introduced the notion of an annihilator graph, and in particular, he proved that if \(R\) is a reduced ring that is not an integral domain, then \(AG(R) = \Gamma(R)\) if and only if \(|Min(R)| = 2\). In this paper, the authors studied when \(AG(R) = \Gamma(R)\) for a non-reduced ring \(R\). Among other things, they showed that \(AG(R) = \Gamma(R)\) which are identical to the join of a complete graph and a null graph if and only if \(\mathrm{ann}_R(Z(R))\) is a prime ideal, if and only if \(R\) has at most two associated primes.
scientific article; zbMATH DE number 6693982