DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Left and right flat modules over an arbitrary ring

Definition

A left $R$ -module $M$ is flat when $- \otimes_{R} M$ is exact on right $R$ -modules; a right $R$ -module $N$ is flat when $N \otimes_{R} -$ is exact on left $R$ -modules.

Depends on

Used by

The flat dimension of a module Definition
A flat nonprojective module Example
Localization is flat and has vanishing positive Tor Example
Flat modules need not have projective dimension zero False statement
Projective left and right modules are flat over an arbitrary ring Lemma
Torsion-free abelian groups are flat Proposition
A left module is flat exactly when Tor one against every right module vanishes Theorem
A right module is flat exactly when Tor one against every left module vanishes Theorem
Over a principal ideal domain flatness is equivalent to torsion-freeness Theorem

Dependency tree · two levels

9 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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)