Coloring of commutative rings

Abstract

The purpose of this article is to present the idea of coloring of a commutative ring. This idea establishes a connection between graph theory and commutative ring theory which hopefully will turn out to be mutually beneficial for these two branches of mathemathics. In this introductory paper we shall mainly be interested in characterizing and discussing the rings which are finitely colorable, leaving aside, for the moment, possible applications to graph theory. Let R be a commutative ring. We consider R as a simple graph whose vertices are the elements of R, such that two different elements x and y are adjacent iff xy = 0. We let x(R) denote the chromatic number of the graph, i.e., the minimal number of colors which can be assigned to the elements of R in such a way that every two adjacent elements have different colors. A subset C = (xi, . . . . x,} is called a clique provided xixj = 0 for all i # j. If R contains a clique with n elements, and every clique has at most y1 elements, we say that the clique number of R is n and write clique R = n. If the sizes of the cliques in R are not bounded we define clique R = co. We shall show that clique R = co actually entails the existence of an infinite clique. Obviously x(R) 3 clique R and for general graph G we certainly may have x(G) > clique G. However, in the case of commutative rings we have not found any example where x(R) > clique R. The lack of such counterexamples together with the fact that we have been able to establish the equality x(R) = clique R for certain (rather wide) classes of rings like reduced and principal ideal rings motivates the following conjecture.