Let $R$ be a commutative ring with identity and let $S$ be a multiplicative subset of $R$. Is it true that for any injective $R$-module like $M$, $S^{-1}M$ (as the $S^{-1}R$-module) is also injective ?



Is it true that if $M$ is injective then $S^{-1}M$ is also injective
No, this is false in general. I quote the Mathematical Review of Dade, Everett C.
*Localization of injective modules*. 
J. Algebra 69 (1981), no. 2, 416–425.



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> Localization of modules over a commutative ring *R* with respect to a multiplicatively closed subset *S* of *R* is an exact functor with a large number of properties, some of which are listed in Theorem 3.76 of J. J. Rotman's book [An introduction to homological algebra, Academic Press, New York, 1979; MR0538169 (80k:18001)]. The fifth property, namely: (LI) the localization $S^{−1}E$ of any injective *R*-module *E* is an injective $S^{−1}R$-module, is false. Two examples are given here showing that arbitrary *R* and *S* need not have the property (LI). Also a positive result is given, showing that (LI) holds for certain non-Noetherian *R* and certain *S*. In particular, if *R* is the polynomial ring $k[x\_1,x\_2,\ldots]$ in a countable number of $x\_n$ over a nonzero Noetherian ring *k*, then (LI) holds for all choices of *S*.
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