This is a homework problem, with a potential problem in the question:
Let $N_1,N_2,\ldots,N_k$ be submodules of M, and let
$i_0,i_1,\ldots,i_l\in\mathbb{Z}^+$
such that
$1=i_0 For each $1\leq j\leq l$, let $P_j=\sum_{r=i_{j-1}+1}^{i_j}{N_r}$. Prove that the sum $\sum_{r=1}^{k}{N_i}$ is direct if and only if the sums $\sum_{j=1}^{l}{P_j}$ and $\sum_{r=i_{j-1}+1}^{i_j}{N_r}$ $(1\leq j \leq l)$ are all direct. // I was thinking there is something wrong with the indices in the question, since the part about "only if the sums ... $\sum_{r=i_{j-1}+1}^{i_j}{N_r}$..." does not seem to even include $N_1$. Any idea about how to correct the question would be greatly appreciated. Sincere thanks.