Graded rings

Graded rings, morphisms and ideals


This note is currently just a "seedling," created initially for quick reference in the context of the tensor algebra construction. I should fill in additional details at some point.

Definition of graded ring

A ring S is called a graded ring if it is (isomorphic to) the direct sum of additive subgroups:

S=S0S1S2

such that SiSjSi+j for all i,j0.

The elements of Sk are said to be homogeneous of degree k, and Sk is called the homogeneous component of S of degree k.

Definition of morphism of graded rings

Suppose S and T are graded rings. A morphism of graded rings is a ring morphism ϕ:ST that respects the graded structures; i.e., satisfies ϕ(Sk)Tk for all k0.

Definition of graded ideal

Suppose S is a graded ring. A graded ideal of S is an ideal I of S such that

I=k=0(ISk).

Examples


  1. The prototypical example of a graded ring is R[x1,,xn], the polynomial ring in n variables over the commutative ring R. Here S0=R corresponds to the constant polynomials, while Sk corresponds to the subgroup of all R-linear combinations of monomials of total degree k.