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 $$i$$.