CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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.

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A ring is reduced exactly when zero is an intersection of primes

Assume the Axiom of Choice.

A commutative ring $R$ is reduced if and only if its zero ideal is the intersection of its prime ideals.

Given: A commutative ring $R$ and the Axiom of Choice.

[L1]

$R$ is reduced exactly when $Nil (R) = (0)$ (The nilradical and reduced rings).

[L2]

The nilradical is the intersection of all prime ideals (The nilradical is the intersection of all prime ideals).

technique · direct

1.1L1L2

If $R$ is reduced, then $Nil (R) = (0)$ by [L1]. Applying [L2] yields $(0) = ⋂_{p \in Spec R} p$ .

1.2L1L2

Conversely, if $(0) = ⋂_{p \in Spec R} p$ , then [L2] shows that $Nil (R) = (0)$ . Now [L1] gives that $R$ is reduced.

Depends on