The zero-divisor graph of a commutative ring. II. (English) Zbl 1035.13004

Anderson, David F.; Frazier, Andrea; Lauve, Aaron; Livingston, Philip S.

Anderson, Daniel D. (ed.) et al., Ideal theoretic methods in commutative algebra. Proceedings of the conference in honor of Professor James A. Huckaba’s retirement, University of Missouri, Columbia, MO, USA. New York, NY: Marcel Dekker (ISBN 0-8247-0553-X/hbk). Lect. Notes Pure Appl. Math. 220, 61-72 (2001).

From the introduction: For part I see D. F. Anderson and P. S. Livingston, J. Algebra 217, 434–447 (1999; Zbl 0941.05062).
Let \(R\) be a commutative ring (with \(1\neq 0)\) and let \(Z(R)\) be its set of zero-divisors. We associate a (simple) graph \(\Gamma(R)\) to \(R\) with vertices \(Z(R)^*= Z(R-\{0\}\), the set of nonzero zero-divisors of \(R\), and for distinct \(x,y\in Z(R)^*\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). Thus \(\Gamma(R)\) is the empty graph if and only if \(R\) is an integral domain. This definition of zero-divisor graph was introduced in part I of this paper. We continue our investigation of the interplay between the ring-theoretic properties of \(R\) and the graph-theoretic properties of \(\Gamma(R)\) begun in part I. In the second section, we give several examples and recall some known results. In the third section, we study cliques (complete subgraphs) of \(\Gamma (R)\) and give an explicit formula for the number of cliques of a given order in \(\Gamma(R)\) when \(R\) is either a finite reduced commutative ring or \(R=\mathbb{Z}_{p^m}\) for \(p\) prime and \(m\geq 2\). In the fourth section, we consider when \(\Gamma(A)\cong\Gamma(B)\) or \(\Gamma (A)=\Gamma(B)\) for commutative rings \(A\) and \(B\). In particular, we show that for finite reduced commutative rings \(A\) and \(B\) which are not fields, \(\Gamma(A)\cong \Gamma(B)\) if and only if \(A\cong B\). The final section contains some miscellaneous comments on \(\operatorname{Aut}(\Gamma (R))\), the group of graph automorphisms of \(\Gamma(R)\), and some cases when \(\Gamma(R)\) is a planar graph.
For the entire collection see [Zbl 0964.00058].

MSC:

13C05	Structure, classification theorems for modules and ideals in commutative rings
13A05	Divisibility and factorizations in commutative rings
05C15	Coloring of graphs and hypergraphs

Keywords:

nonzero zero-divisors; cliques; coloring of rings

Citations:

Zbl 0941.05062