A graph over the commutative rings
Let \(R\) be a commutative ring with identity, and \(M\) be an \(R\)-module. After several graph constructions from commutative rings, there are studies which associate graphs with module. More specifically zero-divisor graph of modules are studied in the recent past. The comaximal graph of a commutative ring \(R\) is the simple undirected graph with vertex set \(R\) and two distinct vertices \(a\) and \(b\) are adjacent if \(Ra+Rb=R\). Motivated by this, the authors of this paper, associate a graph \(G(R,M)\) as the graph with vertex set \(R\) where two distinct elements \(a\) and \(b\) are adjacent if and only if \(aM+bM=M.\) Obviously, \(G(R,R)\) is the comaximal graph of \(R\) and \(G(\{0\},M)\) is the complete graph. Having introduced the graph \(G(R,M),\) the authors considered the relationship between the completeness of subgraphs of \(G(R,M)\) and algebraic properties of \(R\) and \(M\). In addition, they studied the bipartite and complete bipartite subgraphs of the associated graph. Also they have characterized the diameter of certain subgraphs of \(G(R,M)\).
scientific article; zbMATH DE number 6835187
scientific article