CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-30

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A radical ideal omitting a function admits a point that kills the ideal but not the function

Statement

Assume the Axiom of Choice.

Let $k$ be an algebraically closed field, let $A = k [x_{1}, \dots, x_{n}] / I$ be an affine $k$ -algebra, let $J \subseteq A$ be a radical ideal, and let $f \in A ∖ J$ . Then there exists a $k$ -algebra map $φ : A \to k$ such that $J \subseteq \ker φ$ and $φ (f) \neq 0$ .

Facts & Assumptions

Given: The Axiom of Choice, an algebraically closed field $k$ , an affine algebra $A = k [x_{1}, \dots, x_{n}] / I$ , a radical ideal $J \subseteq A$ , and an element $f \notin J$ .

[L1]

In a polynomial ring over an algebraically closed field, $I (V (K)) = \sqrt{K}$ for every ideal $K$ (Strong Nullstellensatz: I(V(I)) equals the radical of I).

[L2]

Maximal ideals of an affine $k$ -algebra are kernels of $k$ -points (Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points).

Proof

technique · contradiction

1.1

Let $π : k [x_{1}, \dots, x_{n}] \to A$ be the quotient map, let $J^{'} = π^{- 1} (J)$ , and choose $\tilde{f} \in k [x_{1}, \dots, x_{n}]$ with $π (\tilde{f}) = f$ . Because $A / J ≅ k [x_{1}, \dots, x_{n}] / J^{'}$ and $J$ is radical, $J^{'}$ is a radical ideal of the polynomial ring.

givenchoose

2.1

Assume, for contradiction, that every $k$ -algebra map $φ : A \to k$ whose kernel contains $J$ also satisfies $φ (f) = 0$ . Then every point $a \in k^{n}$ annihilating $J^{'}$ also annihilates $\tilde{f}$ , so $\tilde{f} \in I (V (J^{'}))$ . By [L1], $I (V (J^{'})) = \sqrt{J^{'}} = J^{'}$ , whence $f = π (\tilde{f}) \in J$ , contradiction.

L1step 1.1assume-contracontradiction

3.1

Therefore some $k$ -algebra map $φ : A \to k$ annihilating $J$ satisfies $φ (f) \neq 0$ . Such a map is a point of the affine algebra by [L2].

L2step 2.1discharge-contradiction∎

Depends on

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Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Theorem (15.7) (standard reference, not scraped)