DefinitionDefinition: Literature-sourcedProof: 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.

Polynomial rings in finitely many commuting indeterminates by iteration

Let $R$ be a commutative ring. Define polynomial rings in finitely many commuting indeterminates recursively by

$R [x_{1}, \dots, x_{0}] : = R, R [x_{1}, \dots, x_{n + 1}] : = R [x_{1}, \dots, x_{n}] [x_{n + 1}] .$

At each stage the coefficient ring embeds as the constant polynomials (Polynomial convolution makes $R [x]$ a commutative ring containing $R$ as its constant subring), so all preceding indeterminates remain present. The new indeterminate commutes with every coefficient by the commutativity built into The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, and consequently all $x_{i}$ commute. This iterated ring is denoted $R [x_{1}, \dots, x_{n}]$ .

Depends on