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The question is very simple: Do you know an easy proof of the following: Let $I$ be an injective $R$-module and $a\subset R$ an ideal. Then the localized module $ I_a$ is again injective.

Of course we are working in commutative algebra. I know a long proof using cohomology with compact suport, where you just have to prove that the kernel of the localization morphism, or the $a$-torsion of $I$, $\Gamma_a(I)=\{m\in I | a^t m=0 \text{ for some } t\in\mathbb{N}\}$ is injective.

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    As I said, $I_a$ is the localized module. Given an ideal $a\in R$, you can localize any $R$-module, just considering the multiplicatively closed system $R-a$. Take a look at Atiyah McDonnald!2012-12-30

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