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.

Filtered modules and the $I$ -adic filtration

Definition

Let $R$ be a commutative ring and let $M$ be an $R$ -module.

A decreasing filtration on $M$ is a sequence of submodules

M = F^{0} M \supseteq F^{1} M \supseteq F^{2} M \supseteq \dots .

If $I \subseteq R$ is an ideal, the $I$ -adic filtration on $M$ is the decreasing filtration

M \supseteq I M \supseteq I^{2} M \supseteq \dots,

where each $I^{n} M$ is the product of the ideal $I^{n}$ with the module $M$ in the sense of The submodule $I M$ generated by products of elements of an ideal $I$ with elements of a module $M$ .

For the ring $R$ itself, viewed as an $R$ -module, this gives the usual $I$ -adic filtration

R \supseteq I \supseteq I^{2} \supseteq \dots .

Depends on

The submodule $IM$ generated by products of elements of an ideal $I$ with elements of a module $M$

Used by

Separated and complete filtered modules Definition
The I-adic topology on a module Definition

Dependency tree · two levels

3 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, §24 (standard reference, not scraped)

Filtered modules and the II-adic filtration