THE CHROMATIC NUMBER OF THE SIMPLE GRAPH ASSOCIATED WITH A COMMUTATIVE RING

抄録

Let R be a Noetherian ring and let Z(R) be the set of all zero-divisors of R. We denote by G(R) the simple graph whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x−y is in Z(R). Let χ(R) be the chromatic number of the graph G(R). If χ(R) is finite, then R is an integral domain or R is a finite Artin ring. In the former case we have χ(R) = 1 and in the latter case we get χ(R) = max{|M1|, . . . , |Mt|} where M1, . . . ,Mt are all maximal ideals of R and |Mi| denotes the number of elements of the set Mi for i = 1, . . . , t.