I-adic filtration ($(I^{n}M)$)

For an ideal $I$, the $I$-adic filtration of a module $M$ is $M\supseteq IM\supseteq I^{2}M\supseteq \cdots$.

If $R$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $I\subseteq \mathfrak{m}$, thenMore generally, the intersection of the powers acts trivially on every finitely generated module.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

For an ideal $I\subseteq R$, the Rees ring is the graded subringIt packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.

The Rees module of an $R$-module $M$ with respect to an ideal $I$ is ${\mathcal{R}}_I(M)={⨁}_{j\ge 0}{I}^{j}M{t}^j$, a graded module over the Rees ring $\mathcal{R}(I)$. Generators of $M$ in degree zero generate this module. For $N\subseteq M$, the graded submodule ${⨁}_j(I^{j}M\cap N)t^j$ is finitely generated when $R$ is a Noetherian ring; a bound on its generators' degrees yields the Artin-Rees lemma.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.