Graded ring

A graded ring is a ring $R$ together with a direct-sum decomposition of abelian groups

(sometimes $n\in\mathbb N$) such that $R_nR_m\subseteq R_{n+m}$ for all $m,n$, and typically $1\in R_0$. The decomposition is an internal direct sum in the category of abelian groups.

Graded rings organize algebra by “degree” and are the ambient objects for graded modules ; a fundamental source is the associated graded ring of a filtration.

Examples:

• The polynomial ring $k[x_1,\dots,x_n]$ is $\mathbb N$-graded by total degree, with $R_d$ the homogeneous polynomials of degree $d$.
• If $R$ has an ideal-adic filtration, the associated graded ring $\bigoplus_{n\ge 0} I^n/I^{n+1}$ is naturally graded.