# Filtered ring

## Definition

A filtered ring is a commutative unital ring $A$ equipped with a filtration, viz., a structure of an ascending chain of subgroups:

$F_0 \subset F_1 \subset F_2 \subset \ldots$

such that the following hold:

• The union of the $F_i$s is $A$
• Each $F_i$ is a subgroup under addition
• $1 \in F_0$
• $F_iF_j \subset F_{i+j}$

It turns out from these that $F_0$ is a unital subring.

## Related notions

• Graded ring: Any graded ring naturally becomes a filtered ring. The filtration associated with the gradation is the filtration where $F_i$ is the sum of the graded pieces from $0$ to $i$.