Rings whose zero-divisor graphs have positive genus
Let \(R\) be a commutative ring with non-zero unity. The concept of the graph of the zero divisors of \(R\) was first introduced by \textit{I. Beck} [see J. Algebra 116, No. 1, 208--226 (1988; Zbl 0654.13001)], where he was mainly interested in colorings. In his work all elements of the ring were vertices of the graph. \textit{D.~F.~Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] associated a graph \(\Gamma(R)\) to \(R\), with vertices \(\mathbb Z(R)\setminus\{0\}\) the set of non-zero zero-divisors of \(R\), and for distinct \(x,y\in\mathbb Z(R)\setminus\{0\}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). In this paper it is shown that for any positive integer \(g\), there are finitely many isomorphism classes of rings whose zero-divisor graph has genus \(g\).
scientific article; zbMATH DE number 5545838