Uniform dimension of modules

Abstract

Let M be a module which has finite uniform dimension and let Ki(1 ≤ i ≤ n) be a finite collection of submodules of M such that 0 = K1 ∩···∩ Kn. Then the uniform dimension u(M) of M is the sum of the uniform dimensions of the factor modules M/Ki(1 ≤ i ≤ n) if and only if Ki is a complement of K1 ∩···∩ Ki−1 ∩ Ki+1 ∩···∩ Kn in M for each 1 ≤ i ≤ n. In case Ki is Pi-prime for some prime ideal Pi for each 1 ≤ i ≤ n, the prime ideals Pi (1 ≤ i ≤ n) are distinct and 0 ≠ K1 ∩···∩ Ki−1 ∩ Ki+1 ∩···∩ Kn for each 1 ≤ i ≤ n, then it is shown that u(M) = ∑i=1nu(Li/(Li ∩ Ki)) for certain submodules Li (1 ≤ i ≤ n) of M.