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I am continuing on my self study in small modules and i have another question: Is it true that $K_1\leq_s M_1 \text{ and } K_2\leq_s M_2\Leftrightarrow K_1\oplus K_2\leq_s M_1\oplus M_2?$

Here the notation $N\leq_s M$ denotes as usual a small module. I'm pretty sure this follows since if $N\leq_sM$ and f:M\to M' is a module morphism then f(N)\leq_s M' which is a fact I've proved. Then my attempt was to work in the two directions using respectively the canonical projections and for the other case the canonical embeddings. In particular I have to convince myself in the case that $K_1\leq_s M_1 \text{ and } K_2\leq_s M_2\Rightarrow K_1\oplus K_2\leq_s M_1\oplus M_2?$ Am I correct working with the embeddings?

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Yes, the embeddings are a good thing to look at.

Note that if $H$ and $K$ are small submodules of $M$, then so is $H+K$: if $(H+K)+L = M$, then $H+(K+L) = M$, so $K+L=M$ by the smallness of $H$; and hence $K+L=M$, so $L=M$ by the smallness of $K$. Thus, $H+K$ is small.

So you just need to verify that $\iota_1(K_1)\leq_{s} M_1\oplus M_2$ and $\iota_2(K_2)\leq_{s}M_1\oplus M_2$.