Filtered ring

This article defines a notion of a ring with additional structure

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
• $1\in F_{0}$
• $F_{i}F_{j}\subset F_{i+j}$

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 $i$.