# Graded ring

From Commalg

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

## Definition

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

such that the following hold:

- Each is a subgroup under addition
- . In other words, if and then

A structure of the above sort on a ring is termed a **gradation**, also a -gradation. The ring is **positively graded** if for all .

There are related notions for noncommutative rings.