$\mathbb{Z}/p^n$ is an injective module over itself.

Question

For any prime power $p^n$, prove that $\mathbb{Z}/p^n$ is an injective module over itself.

Answer 1

More generally, for any PID $R$ and every non-zero element $e \in R$, the ring $R/(e)$ is self-injective:

Baer's criterion implies that, if $S$ is a commutative ring in which every ideal is principal, an $S$-module $M$ is injective if and only if for all $a \in S$, $m \in M$ with $\mathrm{Ann}(a) \subseteq \mathrm{Ann}(m)$ we have $m \in aM$.

This can be checked directly for $M=S=R/(e)$.