I'm a bit new to direct sums etc., so for the purpose of convincing myself, formally, that $(\mathbb{Z} \oplus \mathbb{Z})/\langle (2,2)\rangle = \mathbb{Z} \oplus \mathbb{Z}_2$, using the basis $\{(1,0), (1,1) \}$ for $\mathbb{Z}^2$ I’ve resolved to proving the following claim, and would like to know: (1) if my proof holds; (2) that I’m not mistaken in that I can use it to indeed convince myself of the aforementioned relation and (3) if I’m missing a simpler claim/idea by which the relation can be identified more easily.
Claim:
For $K \oplus L$, if $N$ is normal in $L$ then $(K \oplus L) / N \cong K \oplus (L/N)$.
Proof attempt:
Assume w.l.o.g that $K$’s and $L$’s elements are of the form $(k, 0)$ and $(0, l)$, correspondingly, for $k, l \in \mathbb{Z}$. Then the congruence follows, in fact by equality, since:
$$(k,l)+N = (k,0)+(0,l)+N = (k,0)+ \big((0,l)+N\big)$$
so essentially we get: $(K \oplus L) / N = K \oplus (L/N)$.
Would this be correct? Does it extend to all cases of direct sums?