Beck′s Coloring of a Commutative Ring Original Research Article

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 N1, ..., Ns, such that each A/Ni is finite, then for R = A/Nn11 ··· Nnss, χ(R) = cl(R). Finally, we give a complete listing of all finite rings R with χ(R) ≤ 4.