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We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?

Eric Wofsey
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ramin khosravi
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    This is a little more trivial than Eric Wofsey's response, but if you are asking whether, if $M$ is an injective cogenerator in the category of $R$-modules, then $M_S$ is an injective cogenerator in the category of $R_S$-modules, then the answer is still no, because $\mathbb{Q}/\mathbb{Z} \otimes \mathbb{Q} \cong 0$. – Andrew Dudzik May 19 '16 at 08:11

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Certainly not. For instance, over $\mathbb{Z}$, if you localize any module with respect to the multiplicative set $\mathbb{Z}\setminus\{0\}$, you get a $\mathbb{Q}$-module. A $\mathbb{Q}$-module can never be a cogenerator, since no torsion module can map nontrivially to it.

If you ask only for the localization to be a cogenerator for modules over the localized ring, then as Slade commented, this is still not true. For instance, $\mathbb{Q}/\mathbb{Z}$ is an injective cogenerator over $\mathbb{Z}$, but $\mathbb{Q}/\mathbb{Z}\otimes\mathbb{Q}=0$ is not an injective cogenerator over $\mathbb{Q}$.

Eric Wofsey
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