Extension of scalars

Let $\varphi:R\to S$ be a ring homomorphism of commutative rings, and let $M$ be an R-module .

Definition

The extension of scalars of $M$ along $\varphi$ is the $S$-module

where $\otimes$ is the tensor product . The $S$-module structure is given by multiplication on the first factor:

Extension of scalars is left adjoint to restriction of scalars (equivalently, it is a standard instance of the tensor–Hom adjunction ).

Examples

1. From integers to rationals.
   With $\varphi:\mathbb Z\to\mathbb Q$ and $M=\mathbb Z^n$,

   $$\mathbb Q\otimes_{\mathbb Z}\mathbb Z^n \cong \mathbb Q^n.$$

2. Base change along a quotient.
   For the quotient map $R\to R/I$ (with $I$ an ideal ),

   $$(R/I)\otimes_R M \cong M/IM.$$

3. Localization as extension of scalars.
   For $\varphi:R\to S^{-1}R$ (localization at a multiplicative set $S$),

   $$(S^{-1}R)\otimes_R M \cong S^{-1}M,$$

   i.e. localization of modules is a special case of extension of scalars.