Change of rings

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In algebra, given a ring homomorphism f: R \to S, there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N,

They are related as adjoint functors:

f_! : \text{Mod}_R \leftrightarrows \text{Mod}_S : f^*

and

f^* : \text{Mod}_S \leftrightarrows \text{Mod}_R : f_*.

This is related to Shapiro's lemma.

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