TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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Universal property of $R[x]$ : a coefficient homomorphism and the image of $x$ determine a unique ring homomorphism

Statement

Let $R,S$ be commutative rings, let $\varphi\colon R\to S$ be a unital ring homomorphism, and let $s\in S$ . There is a unique unital ring homomorphism

$\operatorname{ev}_{\varphi,s}\colon R[x]\to S$

that extends $\varphi$ on constant polynomials and sends $x$ to $s$ . It is given by $\operatorname{ev}_{\varphi,s}(\sum_i a_i x^i)=\sum_i\varphi(a_i)s^i$ .

Facts & Assumptions

Given: Commutative rings $R,S$ , a unital ring homomorphism $\varphi\colon R\to S$ , and an element $s\in S$ .

[L1]

Evaluation is the finite sum $f_\varphi(s)=\sum_i\varphi(a_i)s^i$ (Evaluation and roots of a polynomial in a commutative target ring).

[L2]

Polynomial convolution makes $R[x]$ a commutative ring with constant embedding $c\colon R\to R[x]$ (Polynomial convolution makes $R[x]$ a commutative ring containing $R$ as its constant subring).

[L3]

Finitely supported sequences and trimmed coefficient lists have the same coefficients and operations (Finitely supported coefficient sequences and trimmed finite coefficient lists define the same formal polynomials).

[L4]

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

[L5]

Finite sums may be reindexed and iterated 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

The formula in [L1] preserves sums term by term, sends $1$ to $1$ , and sends a convolution product to $\sum_{i,j}\varphi(a_i)\varphi(b_j)s^{i+j}=(\sum_i\varphi(a_i)s^i)(\sum_j\varphi(b_j)s^j)$ by [L5]; thus [L4] makes it a unital ring homomorphism, and [L2] shows that it extends $\varphi$ and sends $x$ to $s$ .

givenL1L2L4L5algebra

2.1

If $\psi\colon R[x]\to S$ is another such homomorphism, [L3] writes every polynomial as a finite sum $\sum_i c(a_i)x^i$ , so [L4] forces $\psi(f)=\sum_i\varphi(a_i)s^i$ ; hence $\psi=\operatorname{ev}_{\varphi,s}$ and uniqueness holds.

step 1.1L2L3L4∎

Depends on

Used by

Cited to discharge well-definedness by Evaluation and roots of a polynomial in a commutative target ring.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 13 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

James McKernan, MIT 18.703 Lecture 21, Lemma 21.3 (standard reference, not scraped)

Universal property of R[x]R[x]R[x]: a coefficient homomorphism and the image of xxx determine a unique ring homomorphism