graded ring

Definition

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

$$R=\underset{i=0}{\overset{\infty }{⨁}}{R}_i$$

where multiplication satisfies $R_i\cdot R_j\subset R_{i+j}$.

In particular $R_0$ is a subring and $1\in R_0$.

Homogeneous decomposition

Any $f\in R$ admits a unique homogeneous decomposition

$$f=\sum _{i=0}^{\infty }{f}_i,f_i\in R_i,f_i=0\text{ if }i\gg 0$$

In this decomposition, $f_i$ is the homogenous component of degree $i$. We call $f$ homogeneous if $f=f_i$ for some $i$. Said another way, $f\in R_i\subset R$.

Homogeneous ideals

An ideal $I\subset R$ is homogeneous if it can be generated by homogeneous elements.

Examples

• Polynomials - the polynomial ring $\mathbb{Z}[x_1,\dots ,x_n]$ is graded by degree.