Beck′s Coloring of a Commutative Ring

Abstract

A commutative ring R can be considered as a simple graph whose vertices are the elements of R and two different elements x and y of R are adjacent if and only if xy = 0. Beck conjectured that χ( R) = cl( R). We give a counterexample where R is a finite local ring with cl( R) = 5 but χ( R) = 6. We show that if A is a regular Noetherian ring with maximal ideals N 1, ..., N s , such that each A/ N i is finite, then for R = A/ N n 1 1 ··· N n s s , χ( R) = cl( R). Finally, we give a complete listing of all finite rings R with χ( R) ≤ 4.