In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups ${\displaystyle R_{i}}$ such that ⁠${\displaystyle R_{i}R_{j}\subseteq R_{i+j}}$⁠. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading.

A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded ⁠${\displaystyle \mathbb {Z} }$⁠-algebra.

The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.

Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article.

A graded ring is a ring that is decomposed into a direct sum

of additive groups, such that

for all nonnegative integers ${\displaystyle m}$ and ⁠${\displaystyle n}$⁠.

A nonzero element of ${\displaystyle R_{n}}$ is said to be homogeneous of degree ⁠${\displaystyle n}$⁠. By definition of a direct sum, every nonzero element ${\displaystyle a}$ of ${\displaystyle R}$ can be uniquely written as a sum ${\displaystyle a=a_{0}+a_{1}+\cdots +a_{n}}$ where each ${\displaystyle a_{i}}$ is either 0 or homogeneous of degree ⁠${\displaystyle i}$⁠. The nonzero ${\displaystyle a_{i}}$ are the homogeneous components of ⁠${\displaystyle a}$⁠.