TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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.

Universal property of a polynomial ring on an arbitrary family of indeterminates

Statement

Let $R, S$ be commutative rings, let $φ : R \to S$ be a ring homomorphism, and let $(s_{i})_{i \in I}$ be a family in $S$ . There is a unique ring homomorphism

$Φ : R [x_{i} : i \in I] ⟶ S$

whose restriction to $R$ is $φ$ and which satisfies $Φ (x_{i}) = s_{i}$ for every $i \in I$ .

Facts & Assumptions

Given: Commutative rings $R, S$ , a ring homomorphism $φ : R \to S$ , and a family $(s_{i})_{i \in I}$ in $S$ .

[L1]

The finite convolution construction is a commutative ring containing $R$ (Finite convolution makes $R [x_{i} : i \in I]$ a commutative ring containing $R$ ).

[L2]

A ring homomorphism preserves addition, multiplication, and the multiplicative identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$ ).

[L3]

Finite sums may be reindexed and evaluated in either order over finite products (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

Proof

technique · direct

1.1

For $a \in M (I)$ define $s^{a} : = \prod_{i \in supp (a)} s_{i}^{a_{i}}$ , and for $c = \sum_{a} c_{a} x^{a}$ define $Φ (c) : = \sum_{a} φ (c_{a}) s^{a}$ ; both expressions are finite and the empty product is $1_{S}$ .

L1construct

2.1

Pointwise addition gives $Φ (c + d) = Φ (c) + Φ (d)$ , while $s^{a + b} = s^{a} s^{b}$ and finite reindexing give $Φ (c d) = Φ (c) Φ (d)$ .

step 1.1L3algebra

3.1

The zero monomial gives $Φ (1) = 1$ , constants give $Φ (r) = φ (r)$ , and the one-supported exponent family gives $Φ (x_{i}) = s_{i}$ ; hence $Φ$ is the required ring homomorphism by [L2].

step 1.1step 2.1L2

4.1

Any ring homomorphism with these values must send $c_{a} x^{a}$ to $φ (c_{a}) s^{a}$ and therefore, by finite additivity, must equal the formula in step 1.1.

step 1.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 12 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

U. Thiel, Commutative Algebra, Section 1.4 (standard reference, not scraped)