Injectivity is a local property

Question

Let $R$ be a commutative noetherian ring, and let $M$ be an $R$-module. How can I show that if any localization $M_p$ at a prime ideal $p$ of the ring $R$ is injective over $R_p$, then $M$ is injective?

Answer 1

Baer's criterion shows that it suffices to show that $\hom(B,M) \to \hom(A,M)$ is surjective for $B=R$ and $A=$ an ideal, in particular both are finitely presented. But then $\hom$ commutes with localization and we are done.