Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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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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Tor does not take two left modules over an arbitrary ring without extra bimodule structure

Statement

False claim: for every noncommutative ring R, Tor⁡iR(M,N) is defined for two left R-modules M,N.

Refutation

Given: the ring R and its left modules.

1.1given

The tensor construction requires its first input to be a right R-module so that (xr)⊗y=x⊗(ry) is meaningful.

2.1step 1.1algebra

For two merely left modules, xr is not part of the supplied structure, so the balancing relation is not typed.

3.1step 2.1algebra∎

Hence the displayed Tor expression is not even defined without additional bimodule or opposite-ring data, refuting the universal claim.

Depends on

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources