Uniform dimension of modules

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