Question about "M" in a short exact sequence

Question

If we have a splitting short exact sequence $0\to L\stackrel {\alpha}\to M\stackrel{\beta}\to N\to 0$ with $r$ as a retraction of $\alpha$. Is it true then that $\alpha(L)\cup \ker(r)=M$?

Answer 1

Take $0\to\mathbb Z\to \mathbb Z\times\mathbb Z\to\mathbb Z\to 0$. Is ($\mathbb Z\times \{0\})\cup(\{0\}\times\mathbb Z)=\mathbb Z\times\mathbb Z$?