DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-08-11

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.

Evaluation and roots of a polynomial in a commutative target ring

Let $φ ⁣ : R \to S$ be a unital ring homomorphism between commutative rings (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ), let $s \in S$ , and let $f = \sum_{i} a_{i} x^{i} \in R [x]$ . The value of $f$ at $s$ along $φ$ is

$f_{φ} (s) : = \sum_{i} φ (a_{i}) s^{i} \in S .$

The sum is finite because the coefficient sequence of $f$ has finite support (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution). When $R$ is identified with a subring of $S$ , the inclusion is understood and the subscript is omitted. An element $s \in S$ is a root or zero of $f$ when $f_{φ} (s) = 0_{S}$ .

Evaluation produces an element of the target ring from a formal polynomial. It does not identify the polynomial with the function that it induces.

Depends on