Does the tensor product distribute over internal direct sums as well?

Question

I have an integral domain $R$, and $R$-modules $M$, $N_1$, $N_2$. I know that there is an $R$- module isomorphism $$M\otimes_R (N_1\oplus N_2)\cong (M\otimes_R N_1)\oplus(M\otimes_R N_2).$$ where $m\otimes (n_1,n_2)\to (m\otimes n_1,m\otimes n_2).$

What is the corresponding isomorphism in the case that $N_1$ and $N_2$ are submodules of $M$ and we regard $N_1\oplus N_2$ as an internal direct sum and how does it follow from the one above?

Answer 1

The external direct sum of $N_1$ and $N_2$ is canonically isomorphic to their internal direct sum by $$N_1\oplus N_2\ \longrightarrow\ N_1+N_2:\ (n_1,n_2)\ \longmapsto\ n_1+n_2,$$ whenever the internal direct sum exists, of course. So nothing changes except for some notation; for the isomorphism you could in stead write $$m\otimes(n_1+n_2)\ \longmapsto\ (m\otimes n_1,m\otimes n_2).$$