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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.

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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.

1 Answers 1

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It should say $0 = i_0 < i_1 < \ldots < i_l = k$. That should correct it, as far as I see.