Graded ring

Last updated January 27, 2022

Definition

Definition. A graded ring is a ring $S$ together with a decomposition $$S=\bigoplus_{d\geq 0}S_d$$ of $S$ into a direct sum of abelian groups $S_d$ such that $$S_d\cdot S_e\subseteq S_{d+e}\quad\forall d,e\geq 0.$$

Homogeneous items

Elements

Definition. An element is homogeneous of degree $d$ if it belongs to $S_d$ for some $d\geq 0$.

Ideals

Definition. An ideal $\mathfrak{a}\subseteq S$ is called homogeneous if $$ \mathfrak{a}=\bigoplus_{d\geq 0}(\mathfrak{a}\cap S_d). $$

Properties

1. Any element in $S$ can be written as a finite sum of homogeneous elements.
2. Let $I$ be a homogeneous ideal of $S$. Then $$S/I=\bigoplus_{j\geq 0}S_j/I_j.$$

Of homogeneous ideals

Let $\mathfrak{a}$ be a homogeneous ideal.

1. An ideal is homogeneous if and only if it can be generated by homogeneous elements.
2. The sum, product, intersection, and radical of homogeneous ideals are homogeneous.
3. To show that $\mathfrak{a}$ is a prime ideal, it suffices to show the condition is satisfied for homogeneous elements
   1. The condition that $fg\in\mathfrak{a}$ implies $f\in\mathfrak{a}$ or $g\in\mathfrak{a}$.

Examples

1. Polynomial ring