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.