Split and non splitting short exact sequence

Question

Please explain that in this,

Why the first sequence is split exact while other is non-split and exact?

Answer 1

A sequence $0\to A\to B\to C\to 0$ is split exact iff $B\cong A\oplus C$ and the given maps correspond to the inclusion $A\to A\oplus C$ and the projection $A\oplus C\to C$ under the isomorphism between them.

Note that $\Bbb Z_{pq} \cong\Bbb Z_p\oplus \Bbb Z_q$, say, because of the Chinese remainder theorem, while $\Bbb Z_{p^2} \not\cong \Bbb Z_p\oplus \Bbb Z_p$, say, because the former contains an element of order $p^2$ while the latter does not.