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.

Uniqueness from squares and cubes

PDF p. 5: Seminormalization in Lemma 2.4

Statement

Let $A$ be a reduced commutative ring. If $a, b \in A$ satisfy $a^{2} = b^{2}$ and $a^{3} = b^{3}$ , then $a = b$ . Here reduced means that the only nilpotent element is zero.

Facts & Assumptions

Given: A reduced commutative ring $A$ and elements $a, b$ with equal squares and equal cubes.

Goal: Prove $a = b$ .

Proof

1.1givenalgebra

From the equalities, $a^{2} b = b^{3} = a^{3}$ and $a b^{2} = a^{3}$ . Consequently $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3} = 0$ .

2.1givenstep 1.1∎

Reducedness gives $a - b = 0$ , so $a = b$ .

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