# Graded rings

## Graded rings, morphisms and ideals

Definition of graded ring

A ring **graded ring** if it is the direct sum of additive subgroups:

such that

The elements of **homogeneous of degree **, and

**homogeneous component of** of degree .

Definition of morphism of graded rings

Suppose **graded ring morphism** is a ring morphism

Definition of graded ideal

Suppose **graded ideal** of

### Examples

- The prototypical example of a graded ring is
, the polynomial ring in variables over the commutative ring . Here corresponds to the constant polynomials, while corresponds to the subgroup of all -linear combinations of monomials of total degree .