The following is an exercise:
Let $r$ and $s$ be positive integers and let $n=rs$. Show that the exact sequence $$0 \to r \mathbb{Z}_n \stackrel{f}{\to}\mathbb{Z}_n \stackrel{g}{\to} s\mathbb{Z}_n \to 0$$ splits if and only if $gcd(r,s) = 1$. Here $f$ is the inclusion and $g(\overline{x}) = s \overline{x}$.
I already showed that $gcd(r,s) = 1$ implies the splitting of the sequence. I am having trouble with the converse.
If the sequence splits, then there exists an isomophism $\varphi: r\mathbb{Z}_n \oplus s\mathbb{Z}_n \to \mathbb{Z}_n$ satisfying
$$\varphi(r \overline{x},0) = r\overline{x}$$ and $$s\varphi(r\overline{x}, s\overline{y}) = s\overline{y}$$ for every $\overline{x}, \overline{y} \in \mathbb{Z}_n$. My aim is to prove the existence of a pair of integers $\alpha$ and $\beta$ such that $$\alpha r + \beta s = 1.$$
I don't know how to proceed.