If $M\bigoplus N $ submodule $A\bigoplus B$ does it imply either $M$ submodule $A$ or $M$ submodule $ B$?

Question

If $M\bigoplus N$ is a submodule of $A\bigoplus B$ does it necessarily imply either $M$ is a submodule of $A$ or $M$ is a submodule of $B$?

If in general it is not true, is it true if $M$ and $N$ are $\mathbb Z$-modules?

I tried finding counter examples when they are $\mathbb Z$-modules but did not succeed.

Answer 1

No,

Let $A,B = \mathbb{Z}$ as $\mathbb{Z}$-modules. Let $M$ be the submodule of $A\oplus B$ generated by $(2,1)$ and let $N$ be the submodule generated by $(0,1).$ Then $M\oplus N \subset A\oplus B$ but $M$ is not a subset of $A$ or $B.$

Answer 2

No, in either cases really.

Consider $V=\bigoplus_{p\in\Bbb P}\Bbb Z/p\Bbb Z$ (where $\Bbb P$ is all the prime integers). Now partition $\Bbb P$ into $A,B$ and $A',B'$ such that neither $A\subseteq A'$ nor $B\subseteq B'$, and consider the decompositions defined by $A,B$ and $A',B'$ as counterexamples.