The \(M\)-principal graph of a commutative ring. (English) Zbl 1340.05125
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 \(n > 1\).
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 \(n > 1\).
Reviewer: Vilmar Trevisan (Porto Alegre)
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C69 | Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) |
| 13A99 | General commutative ring theory |
| 13C99 | Theory of modules and ideals in commutative rings |
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