Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03
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.

Annihilators, torsion elements and the torsion subset of a module

Definition

For a left RR-module MM and m∈Mm\in M, the annihilator of mm is

Ann⁡R(m):={r∈R:rm=0M},\operatorname{Ann}_R(m):=\{r\in R:rm=0_M\},

and the annihilator of MM is

Ann⁡R(M):={r∈R:rm=0M for every m∈M}.\operatorname{Ann}_R(M):=\{r\in R:rm=0_M\text{ for every }m\in M\}.

If RR is an integral domain, an element m∈Mm\in M is a torsion element when rm=0Mrm=0_M for some nonzero r∈Rr\in R. The set of all torsion elements is denoted Tor⁡(M)\operatorname{Tor}(M); MM is torsion-free when Tor⁡(M)={0M}\operatorname{Tor}(M)=\{0_M\}.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

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Sources