Definition 1

Let $A$ be a commutative ring, $m$ an ideal of $A$, $E$ an $A$-module and $(E_n)$ a filtration on the additive group $E$ consisting of submodules of $E$. The filtrations $(E_n)$ is called m-good if: (1) $m E_n \subset E_{n+1}$ for all $n \in \mathbf{Z}$; (2) there exists an integer $n_0$ such that $m E_n = E_{n+1}$ for $n \geq n_0$.

Then by induction on $q$, $m^q E_n = E_{n+q}$ for $n \geq n_0, q \geq 1$. Note that condition (1) means that the filtration $(E_n)$ is compatible with the $A$-module structure on $E$ if $A$ is given the $m$-adic filtration. Clearly, on every $A$-module $E$, the $m$-adic filtration is m-good. If a filtration on an $A$-module $E$ is m-good, the quotient filtration on any quotient module of $E$ is m-good.