Filtered ring

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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_is 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.