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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $1_{S} \otimes u : S \otimes_{R} M \to S \otimes_{R} M^{'}$ .

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

Definition

Depends on

Dependency tree · 2 levels

Sources

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

Definition

Depends on

Dependency tree · 2 levels

Sources

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