The \(M\)-principal graph of a commutative ring (Q397091): Difference between revisions

In the present paper, the authors introduce the \(M\)-principal graph of \(R\), denoted by \(M - \mathrm{PG}(R)\), where \(M\) is an \(R\)-module of a commutative ring \(R\). It is the graph whose vertex set is \(R\setminus\{0\}\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(x M = y M\). They study properties of \(\mathrm{PG}(R)\) and some relations between \(\mathrm{PG}(R)\), when \(R=M\), and \(M - \mathrm{PG}(R)\) are established. In particular, the authors consider the graph \(\mathrm{PG}(\mathbb{Z}_n)\) for each positive integer \(n > 1\).