A Note on Graphical Representation of Rings

Abstract

Let R be a commutative ring with identity. Let G be a graph with vertices as elements of R, where two distinct vertices x and y are adjacent if and only if Rx + Ry = R. In this paper we show that a commutative ring R is a finite ring if and only if the graph G (associated with R as above) is finitely colourable. Moreover we show that in this case the chromatic number of the graph G is the sum of the number of maximal ideals and the number of units of R. © 1995 Academic Press, Inc.