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.

Primewise detection of torsion

PDF p. 5: Proposition 2.6: torsion freeness

Statement

An abelian group $M$ is torsion-free if multiplication by every prime is injective.

Facts & Assumptions

[F1]

Torsion means annihilation by some positive integer, as in Annihilators, torsion elements and the torsion subset of a module.

Proof

Given: $M$ satisfying the stated injectivity.

Goal: Show every torsion element is zero.

1.1given

For $n x = 0$ with $n > 0$ , factor $n$ into primes. Multiplication by $n$ is the composition of the corresponding injective prime multiplication maps, hence is injective (the empty composition for $n = 1$ is the identity).

2.1F1step 1.1∎

Thus $n x = 0$ implies $x = 0$ , as required.

Depends on

Annihilators, torsion elements and the torsion subset of a module

Used by · 2

Dependency tree · 3 levels

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