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