Filtration of a module

Builds on Commutative ring Module over a ring Submodule Ideal

Definition

Let $R$ be a commutative ring, let $M$ be an $R$ -module, and let $I$ be an ideal of $R$ . A filtration of $M$ is a sequence of submodules $(M_n)_{n \ge 0}$ with $M_0 = M$ and $M_n \supseteq M_{n+1}$ for all $n$ . It is an $I$ -filtration if moreover $I M_n \subseteq M_{n+1}$ for all $n \ge 0$ . An $I$ -filtration is stable if there is $n_0$ such that $I M_n = M_{n+1}$ for all $n \ge n_0$ .

Remarks

Example: the $I$ -adic filtration $M_n = I^n M$ is a stable $I$ -filtration, since $I \cdot I^n M = I^{n+1} M$ for every $n$ . Non-example: for $R = M = \Z$ and $I = 2\Z$ , the filtration $M_n = 4^n \Z$ is not a $2\Z$ -filtration, since $I M_1 = 8\Z$ is not contained in $M_2 = 16\Z$ . Some authors say $I$ -good for stable. Two stable $I$ -filtrations of $M$ have bounded difference: there is $k$ with $M_{n+k} \subseteq M'_n$ and $M'_{n+k} \subseteq M_n$ for all $n$ , so they define the same topology.

Reading path

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Commutative ring concept
Ideal concept
Module over a ring concept
Submodule concept
Filtration of a module definition

Tag	`GEJYZEY7Y0`
URI	`m2://GEJYZEY7Y0`
Kind	definition
Writing	W0 stub
MSC	13A30, 13C05
Defines	Filtration of a module
Uses	Commutative ring, Module over a ring, Submodule, Ideal
Changed	2 November 2026

Filtration of a module

Definition

Remarks

Reading path

Further reading

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