graded ring

Definition

A graded ring is a ring with a decomposition as abelian groups

where multiplication satisfies .

In particular is a subring and .

Homogeneous decomposition

Any admits a unique homogeneous decomposition

In this decomposition, is the homogenous component of degree . We call homogeneous if for some . Said another way, .

Homogeneous ideals

An ideal is homogeneous if it can be generated by homogeneous elements.

Examples

• Polynomials - the polynomial ring is graded by degree.