Extension of scalars

In algebra, given a ring homomorphism ${\displaystyle f:R\to S}$, 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:

and

This is related to Shapiro's lemma.

Restriction of scalars changes S-modules into R-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction.

Let ${\displaystyle R}$ and ${\displaystyle S}$ be two rings (they may or may not be commutative, or contain an identity), and let ${\displaystyle f:R\to S}$ be a homomorphism. Suppose that ${\displaystyle M}$ is a module over ${\displaystyle S}$. Then it can be regarded as a module over ${\displaystyle R}$, if the action of ${\displaystyle R}$ is given via ${\displaystyle r\cdot m=f(r)\cdot m}$ for ${\displaystyle r\in R}$ and ${\displaystyle m\in M}$.

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