DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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
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
Extension of scalars carries flat modules to flat modules Proposition
Flatness is transitive under a flat change of rings Proposition
Extension of scalars is left adjoint to restriction of scalars Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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