This article defines a notion of a ring with additional structure
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.
- 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 .