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

Dependency tree · 2 levels

Sources