Definition 2

Given a ring $A$, a filtration $(A_n)_{n \in \mathbf{Z}}$ over the additive group $A$ is called compatible with the ring structure on $A$ if

(1)

(2)

The ring $A$ with this filtration is then called a filtered ring.

Conditions (1) and (2) show that $A_n$ is a subring of $A$ and the $A_n$ (left and right) $A_0$-modules. The set $B = \bigcup_{n \in \mathbf{Z}} A_n$ is a subring of $A$ and the set $n = \bigcap_{n \in \mathbf{Z}} A_n$, a two-sided ideal of $B$; for if $x \in n$ and $a \in A_k$ for all $k \in \mathbf{Z}$, $x \in A_{k-p}$, whence $ax \in A$, and $xa \in A$, by (1); therefore $ax \in n$ and $xa \in n$.

An important particular case is that in which $A_n = A$; then $A_n = A$ for $n \leq 0$ and all the $A_n$ are two-sided ideals of $A$.