Finite rings with complete bipartite zero-divisor graphs. (Q1759249)

In this paper, a zero-divisor graph \(\Gamma(R)\) has the vertex set consisting of all one-sided and two sided zero-divisors, and two distinct vertices \(x\) and \(y\) are adjacent if and only if either \(xy=0\) or \(yx=0\). The authors classify all finite associative (not necessarily commutative or containing an identity element) rings whose zero-divisor graphs \(\Gamma(R)\) are complete bipartite. Among the thirteen non-isomorphic classes of rings, nine are specific finite rings with rather small cardinalities (e.g., all the corresponding commutative rings with 1 are \(\mathbb Z_8\), \(\mathbb Z_9\), \(\mathbb Z_3[x]/(x^2)\), \(\mathbb Z_2[x]/(x^3)\) and \(\mathbb Z_4[x]/(2x,x^2)\)); while each of the other four remaining classes of rings is a direct product of two rings involving a finite field (e.g., \(\mathrm{GF}(q_1)\times\mathrm{GF}(q_2)\), \(\mathrm{GF}(q)\times\mathbb Z_4\), \(\mathrm{GF}(q)\times\mathbb Z_2[x]/(x^2)\)).