Filtration of a module

A decreasing filtration of a module is a chain $M=M_0\supseteq M_1\supseteq \cdots$ of submodules.

Two decreasing filtrations $(M_n)$ and $(N_n)$ are equivalent when each contains a fixed shift of the other: $M_{n+a}\subseteq N_n$ and $N_{n+b}\subseteq M_n$ for fixed $a,b$ and every $n$.

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

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.

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

A filtered ring is a ring with subgroups $(R_n)$ satisfying $R_m{R}_n\subseteq R_{m+n}$.