, Volume 67, Issue 2, pp 211-220
Date: 16 Aug 2013

The regular graph of a commutative ring

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Abstract

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+yZ(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2 n , where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.

Communicated by László Fuchs