Why we have $N_{1}+N_{2}+\cdots+N_{n}=N_{1}\oplus N_{2}\oplus\cdots\oplus N_{n}$?

Question

Let $R$-module $M$ be a direct sum of modules $M_{1},M_{2},...,M_{n}$ and for each $i=\overline{1,n}$, $N_{i}\leq M_{i}$ ($N_{i}$ is submodule of $M_{i}$). Why we have $N_{1}+N_{2}+\cdots+N_{n}=N_{1}\oplus N_{2}\oplus\cdots\oplus N_{n}$ and $M/\left(\oplus_{i=1}^{n}N_{i}\right)\cong\oplus_{i=1}^{n}\left(M_{i}/N_{i}\right)$?

Answer 1

Let's denote by $\times$ the “external” direct sum just to avoid confusion. Saying that $$ N_1+N_2+\dots+N_n=N_1\oplus N_2\oplus\dots\oplus N_n $$ means that the obvious homomorphism $$ N_1\times N_2\times\dots\times N_n\to N_1+N_2+\dots+N_n $$ is injective.

Hint for the second part: you can define homomorphisms $$ M_i\to M\big/\left(\textstyle\bigoplus_{i=1}^{n}N_{i}\right) \quad(i=1,2,\dots,n) $$ in a very natural way. What are their kernels?