Direct sum of isomorphic modules

Question

Suppose $M$ is a module and let $M', M''$ be submodules of $M$.

Suppose there are modules $N \cong M'$ and $N' \cong M''$.

Under what conditions is $N\oplus N' \cong M' \oplus M'' \leq M$?

Answer 1

$M'+ M''=M'\oplus M''$ is direct (as a sum of these two submodules) iff $M'\cap M''=\{0\}$.

$N\oplus N'\cong M'\oplus M''$ always (as external direct sums) using the obvious isomorphism built out of the ones between $N\cong M'$ and $N'\cong M''$. Here, it does not matter whether or not $M'\cap M''=\{0\}$ or not.

So the answer to your question seems to be just "When $M'\cap M''=\{0\}$."