On endomorphism-regularity of zero-divisor graphs (Q942115)

Let \(G\) be a graph and \(\text{End}(G)\) the semigroup consisting of all the endomorphisms of \(G\). An element \(a\) of a semigroup \(S\) is called regular if \(a=aba\) for some \(b\in S\), and \(S\) is called regular if every element in \(S\) is regular. A graph \(G\) is called end-regular if \(\text{End}(G)\) is regular. The zero-divisor graph of a ring \(R\), denoted by \(\Gamma(R)\), is a graph whose vertex set \(Z(R)^\ast\) consisting of all non-zero zero-divisors of \(R\) and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). Let \(R\) be a ring containing nontrivial idempotents. Then the main result of the present paper is to prove that \(\Gamma(R)\) is end-regular if and only if \(R\) is isomorphic to one of the following rings: \(\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\); \(\mathbb{Z}_2\times\mathbb{Z}_4\); \(\mathbb{Z}_2\times(\mathbb{Z}_2[x]/(x^2))\); \(F_1\times F_2\), where \(F_1\), \(F_2\) are fields.