# Filtered ring

From Commalg

*This article defines a notion of a ring with additional structure*

## Definition

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

such that the following hold:

- The union of the s is
- Each is a subgroup under addition

It turns out from these that 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 is the sum of the graded pieces from to .