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 = S_{0} \oplus S_{1} \oplus S_{2} \oplus \dots

such that $S_{i} S_{j} \subseteq S_{i + j}$ for all $i, j \geq 0$ .

The elements of $S_{k}$ are said to be homogeneous of degree $k$ , and $S_{k}$ 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 $ϕ : S \to T$ that respects the graded structures; i.e., satisfies $ϕ (S_{k}) \subseteq T_{k}$ for all $k \geq 0$ .

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}^{\infty} (I \cap S_{k}) .

Examples

The prototypical example of a graded ring is $R [x_{1}, \dots, x_{n}]$ , the polynomial ring in $n$ variables over the commutative ring $R$ . Here $S_{0} = R$ corresponds to the constant polynomials, while $S_{k}$ corresponds to the subgroup of all $R$ -linear combinations of monomials of total degree $k$ .