LemmaStatement: AI-generatedProof: AI-generated

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.

Intermediate rings in finite algebras

PDF p. 5: Seminormalization in Lemma 2.4

Statement

Let $A$ be a noetherian commutative ring and $B$ a finite $A$ -algebra. Any $A$ -subalgebra $C \subseteq B$ is finite as an $A$ -module. Here noetherian means every ideal is finitely generated, and finite means finitely generated as a module.

Facts & Assumptions

Given: A noetherian ring $A$ , a finitely generated $A$ -module $B$ with an $A$ -algebra structure, and an intermediate $A$ -subalgebra $C$ .

Goal: Show $C$ is finitely generated over $A$ .

Proof

1.1givenconstruct

A submodule of $A^{n}$ is finitely generated: induct on $n$ , projecting to the last coordinate; the image is a finitely generated ideal and the kernel is a submodule of $A^{n - 1}$ . Lifts of generators of the image, together with generators of the kernel, generate the submodule.

2.1givenstep 1.1construct∎

Choose a surjection $A^{n} \to B$ . The inverse image of $C$ is finitely generated by step 1.1, and its generators map to generators of $C$ . Hence $C$ is finite over $A$ .

Used by · 1

Seminormality and seminormalization