If we're given an arbitrary principal ideal domain $R$ and some $R$-module, call it $M$, is there an example of an $M$ that is not injective while $M_x$ is an injective module $\forall x \in max(R)$? Further is there an example of an injective M, while $M_x$ is not an injective module?
Or do we always have that one implies the other?