# Change of rings

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,

• $f_! M = M \otimes_R S$, the induced module.
• $f_* M = \operatorname{Hom}_R(S, M)$, the coinduced module.
• $f^* N = N_R$, the restriction of scalars.

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.