DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

The $I$ -adic completion of a module

Definition

Let $R$ be a commutative ring, let $I \subseteq R$ be an ideal, and let $M$ be an $R$ -module.

The $I$ -adic completion of $M$ is the inverse limit ${\hat{M}}^{I} : = {\underset{\leftarrow}{l i m}}_{n \geq 1} M / I^{n} M$ of the quotient system $M / I M \leftarrow M / I^{2} M \leftarrow M / I^{3} M \leftarrow \dots .$

When the ideal $I$ is fixed, this module is denoted simply by $\hat{M}$ .

The associated completion map is $κ_{M} ⁣ : M \to \hat{M}, m \mapsto (m m o d I^{n} M)_{n \geq 1} .$

The $I$ -adic completion of a module

Definition

Depends on

Dependency tree · 2 levels

Sources

The II-adic completion of a module

Definition

Depends on

Dependency tree · 2 levels

Sources

The $I$ -adic completion of a module