Graded ring

This article defines a notion of a ring with additional structure

Definition

A graded ring is a commutative unital ring $A$ equipped with a direct sum decomposition as a sum of Abelian subgroups:

$A = \oplus_{i = - \infty}^{\infty} A_{i} = \dots A_{- 2} \oplus A_{- 1} \oplus A_{0} \oplus A_{1} \oplus A_{2} \oplus \dots$

such that the following hold:

Each $A_{i}$ is a subgroup under addition
$1 \in A_{0}$
$A_{m} A_{n} \subset A_{m + n}$ . In other words, if $a \in A_{m}$ and $b \in A_{n}$ then $a b \in A_{m + n}$

A structure of the above sort on a ring is termed a gradation, also a $Z$ -gradation. The ring $A$ is positively graded if $A_{i} = 0$ for all $i < 0$ .

There are related notions for noncommutative rings.

Related notions

Weaker notions

Filtered ring