DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

Algebras over a commutative ring, central structure maps, and algebra homomorphisms

Definition

Let $R$ be a commutative ring (Commutative ring). An $R$ -algebra is a unital ring $A$ together with a unital ring homomorphism

$η_{A} : R ⟶ A$

(Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ) whose image is central: $η_{A} (r) a = a η_{A} (r)$ for every $r \in R$ and $a \in A$ . The induced scalar action is $r a : = η_{A} (r) a$ , making $A$ an $R$ -module and multiplication $A \times A \to A$ bilinear.

An $R$ -algebra homomorphism $f : A \to B$ is a unital ring homomorphism satisfying $f \circ η_{A} = η_{B}$ . Such a map is automatically $R$ -linear. An $R$ -algebra is commutative when its underlying ring is commutative.

Depends on

Used by

The tensor product of R-algebras has multiplication (a⊗ b)(a'⊗ b')=aa'⊗ bb' Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 13 results over 11 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

W. Li, Commutative Algebra, Lectures 9-10 (standard reference, not scraped)