Graphs associated to co-maximal ideals of commutative rings
Let \(R\) be a commutative ring with \(1\). \textit{P. K. Sharma} and \textit{S. M. Bhatwadekar} [J. Algebra 176, 124--127 (1995; Zbl 0838.05051)] defined a graph on \(R\), \( \Gamma(R)\), with vertices as elements of \(R\), where two distinct vertices \(a\) and \(b\) are adjacent if and only if \(aR + bR = R\). Let \(\Gamma_{2}(R)\) be the subgraph of \(\Gamma(R)\) induced by non-unit elements. In this paper, the author derives several properties of \(\Gamma_{2}(R)\). He was able to characterize those rings \(R\) for which \(\Gamma_{2}(R)-J(R)\) is a forest and those rings \(R\) for which \(\Gamma_{2}(R)-J(R)\) is Eulerian, where \(J(R)\) is the Jacobson radical of \(R\). He was also able to show that \(\omega(\Gamma_{2}(R)) = | \text{Max}(R)| \), where \(\omega(G)\) is the clique number of a graph \(G\) and \(\text{Max}(R)\) is the set of maximal ideals of \(R\). Moreover, he found all finite rings \(R\) such that the genus of \(\Gamma_{2}(R)\)(respectively \(\Gamma(R)\)) is at most one. Moreover the author was able to find all finite rings \(R\) such that \(\Gamma(R)\)(respectively \(\Gamma_{2}(R)\)) is either planar or toroidal.
scientific article; zbMATH DE number 5376939
scientific article