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On the associated graphs to a commutative ring. (English) Zbl 1238.13015

Summary: Let \(R\) be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset \(S\) of \(R\), we associate a simple graph denoted by \(\Gamma _{S}(R)\) with all elements of \(R\) as vertices, and two distinct vertices \(x, y \in R\) are adjacent if and only if \(x+y \in S\). Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of \(\Gamma _{S}(R)\). Moreover, we will improve and generalize some results for the total and the unit graphs.

MSC:

13A99 General commutative ring theory
05C75 Structural characterization of families of graphs
13A15 Ideals and multiplicative ideal theory in commutative rings
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