All submodules of direct sum of modules?

Question

Let $M_1$ and $M_2$ be submodules. Is it true that all submodules of $M_1 + M_2$ are given by $N_1 + N_2$ where $N_1$ and $N_2$ are submodules of $M_1, M_2$. What about in the infinite case? $$\bigoplus M_i$$ What do all the submodules look like?

Answer 1

It is not true. Consider a vector space $k\oplus k$ over a field $k$. This is two dimensional vector space over $k$. It has plenty one-dimensional subspaces (lines through zero). In particular, more than just two following $$k\oplus 0, 0\oplus k$$

Note that even if $k = \mathbb{Z}/2\mathbb{Z}$, then inside $\mathbb{Z}/2\mathbb{}\oplus \mathbb{Z}/2\mathbb{Z}$ there is a subspace generated by $(1,1)$, which is not the one from the description in your question. For larger fields, there are even more lines.