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Can one do it?

I'm trying to prove that $S^{-1}I$ is an injective $S^{-1}R$-module whenever $I$ is an injective $R$-module.

So I need to start with a situation where I have:

(i) $S^{-1}R$-modules $M,N$. (ii) a homomorphism $j:M\to S^{-1}I$ (iii) an injective homomorphism $i:M\to N$.

From this I want to strip the situation down, (by embedding $I\to S^{-1}I$).

I almost have a solution but I think it comes down to being able to interpret $M$ as an $R$-module. Can this be done or do I have to remove elements of $M$ to make it work?

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    You have a canonical map $\varphi$ from $R$ to $S^{-1}R$ that allows you to view an $S^{-1}R$-module as an $R$-module. The action of $r\in R$ is given by the action of $\varphi(r).$2012-02-07

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