DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 $R$ -module $M$ and $m \in M$ , the annihilator of $m$ is

${Ann}_{R} (m) : = {r \in R : r m = 0_{M}},$

and the annihilator of $M$ is

${Ann}_{R} (M) : = {r \in R : r m = 0_{M} for every m \in M} .$

If $R$ is an integral domain, an element $m \in M$ is a torsion element when $r m = 0_{M}$ for some nonzero $r \in R$ . The set of all torsion elements is denoted $Tor (M)$ ; $M$ is torsion-free when $Tor (M) = {0_{M}}$ .

Depends on

Dependency tree · 2 levels

Sources

McGerty, Algebra II: Rings and Modules, Section 3