Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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  • 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.
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Tor is not symmetric as a typed expression over every noncommutative ring

Statement

False claim: Tor is a symmetric bifunctor of two left modules over every ring.

Refutation

Given: a noncommutative ring R and two left modules.

1.1given

The ordinary tensor product M⊗RN already requires one factor to be right-handed.

2.1step 1.1algebra

Thus the supposed inputs of the symmetric expression are not generally a valid domain for Tor.

3.1step 2.1algebra∎

Commutativity supplies an identification of left and right actions; without it the asserted symmetry is ill-typed, not a theorem.

Depends on

Used by

Nothing in the library uses this result yet.

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Sources