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 topology on 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 topology on $M$ is the linear topology whose distinguished neighbourhood basis of $0$ is

{I^{n} M : n \geq 0} .

Equivalently, a subset $U \subseteq M$ is open when for every $x \in U$ there exists $n$ such that

x + I^{n} M \subseteq U .

Thus the basic open neighbourhoods of an arbitrary point $x \in M$ are the cosets

x + I^{n} M (n \geq 0) .

Depends on

Filtered modules and the $I$-adic filtration

Used by

The I-adic completion of a module Definition
Equivalent adic filtrations have canonically isomorphic completions Example
Kernel and universal property of adic completion Theorem

Dependency tree · two levels

2 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

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., §22.1 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, Lemma 24.1 (standard reference, not scraped)

The II-adic topology on a module