DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-16

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Restriction of scalars and extension of scalars $S \otimes_{R} M$ along a ring homomorphism $R \to S$

Definition

Let $f : R \to S$ be a unital homomorphism of commutative rings (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

For an $S$ -module $N$ , its restriction of scalars along $f$ , denoted ${Res}_{R}^{S} N$ , is the same abelian group with $R$ -action

$r \cdot n : = f (r) n .$

For an $R$ -module $M$ , its extension of scalars along $f$ is

$S \otimes_{R} M .$

Here $S$ is an $(S, R)$ -bimodule with left action by multiplication and right action $s \cdot r : = s f (r)$ ( $(S, R)$ -bimodules and commuting left and right scalar actions). The induced outer action

$s^{'} (s \otimes m) : = (s^{'} s) \otimes m$

makes $S \otimes_{R} M$ an $S$ -module (A commuting outer scalar action descends to a tensor product).

Restriction acts on a homomorphism by leaving its underlying function unchanged. Extension sends $u : M \to M^{'}$ to $1_{S} \otimes u : S \otimes_{R} M \to S \otimes_{R} M^{'}$ .

Depends on

Used by

Change of rings: N⊗_RM≅ N⊗_S(S⊗_RM) Corollary
The derived algebra of a solvable Lie algebra is nilpotent in characteristic zero Corollary
Galois conjugates of a representation Definition
Restriction, induction, and coinduction Definition
Semilinear Galois actions, twists, and split central idempotents Definition
For a field extension K/F, one has K⊗_FFⁿ≅ Kⁿ Example
For a field extension K/F, one has K⊗_FMₙ(F)≅ Mₙ(K) as K-algebras Example
S⊗_RR[x]≅ S[x] as S-algebras Example
Base change for intertwiner spaces Lemma
Universal finite projective cohomology complex over any base Lemma
Extension of scalars carries flat modules to flat modules Proposition
Extension to the fraction field recovers the free rank of a finitely generated PID module Proposition
Flatness is transitive under a flat change of rings Proposition
Existence and characteristicity of the nilradical in characteristic zero Theorem
Extension of scalars is left adjoint to restriction of scalars Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

MIT 18.721 Algebraic Geometry notes, Section 2.1 (standard reference, not scraped)
W. Li, Commutative Algebra, Lectures 9-10 (standard reference, not scraped)

Restriction of scalars and extension of scalars S⊗RMS⊗R​M along a ring homomorphism R→SR→S