THE DIMENSION GRAPH FOR MODULES OVER COMMUTATIVE RINGS

Let R be a commutative ring and M be an R-module. The dimension graph of M, denoted by DG(M), is a simple undirected graph whose vertex set is Z(M) \ Ann(M) and two distinct vertices x and y are adjacent if and only if dimM/(x, y)M = min{dimM/xM, dimM/yM}. It is shown that DG(M) is a disconnected graph if and only if (i) Ass(M) = {p, q}, Z(M) = p. q and Ann(M) = p boolean AND q. (ii) dimM = dimR/p = dimR/q. (iii) dimM/xM = dimM for all x is an element of Z(M) \ Ann(M). Furthermore, it is shown that diam(DG(M)) <= 2 and gr(DG(M)) = 3, whenever M is Noetherian with vertical bar Z(M) \ Ann(M)vertical bar >= 3 and DG(M) is a connected graph.