It is well-known that the ring of formal power series $F[[t_1, ..., t_n]]$, where $F$ is a field of characteristic zero, is an injective abelian group. Is it true however that $F[[t_1, ..., t_n]]$ is an injective module over itself? It doesn't seem to be that clear.
While the functor $\mathrm{Hom}_{\mathbb Z}(-, F[[t_1, ..., t_n]])$ is clearly exact, it is not so clear if $\mathrm{Hom}_{F[[t_1, ..., t_n]]}(-, F[[t_1, ..., t_n]])$ is exact.
An injective module over an integral domain is divisible. The power series ring is not a divisible module over itself (multiplication by $t_1$ is not surjective). This holds for $D[[t_1,\dots,t_n]]$, where $D$ is any integral domain of any characteristic.
More generally still, a self-injective integral domain must be a field.