The regular graph of a commutative ring
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THE REGULAR GRAPH OF A COMMUTATIVE RING S. Akbari1 and F. Heydari2 1
Department of Mathematical Sciences, Sharif University of Technology Tehran, Iran E-mail: : s [email protected] 2
Department of Mathematics, Karaj Branch, Islamic Azad University Karaj, Iran E-mail: [email protected] (Received November 6, 2011; Accepted July 1, 2012) [Communicated by L´ aszl´ o Fuchs]
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 + y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 6∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n , 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.
1. Introduction Let R be a commutative ring, let Z(R) be the set of zero-divisors of R and Reg(R) = R\Z(R). Throughout the paper we assume that R is a commutative ring with unity. The regular graph of R, G(R), was first introduced in [2]. It is a graph whose vertex set is Reg(R) and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Here we denote the intersection of all prime ideals and the intersection of all maximal ideals of R by N (R) and J(R), respectively. Thus N (R) is the set of all nilpotent elements of R and J(R) is the Jacobson radical of R. A ring R with N (R) = 0 is called reduced. We denote the characteristic of R by char R. A ring having just one maximal ideal is called a local ring and a ring having only finitely many maximal ideals is said to be a semilocal ring. The Mathematics subject classification numbers: 05C15, 05C25, 05C69, 13A, 13E05. Key words and phrases: regular graph, Noetherian ring, zero-divisors, clique number, chromatic number. 0031-5303/2013/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
S. AKBARI and F. HEYDARI
Q direct product of a family of rings {Ri | i ∈ I} is denoted by i∈I Ri . For an R-module M , the idealization of M over R is the commutative ring formed from R × M by defining addition and multiplication as (r, m) + (s, n) = (r + s, m + n) and (r, m)(s, n) = (rs, rn + sm), respectively. A standard notation for this ring is R(+)M . For basic properties of rings resulting from the idealization construction, the interested reader is referred to [5]. Let G be a graph. The set of vertices and the set of edges of G is denoted by V (G) and E(G), respectively. We say that G is connected if there is a path between any two distinct vertices of G. A tree is a connected graph with no cycle. A forest is a graph whose every component is a tree. A graph in which each pair of distinct vertices is joined by an edge is called a complete graph. We denote the complete graph on n vertices by Kn . A clique in G is a set of pairwise adjacent vertice
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