Graded ring

Builds on Commutative ring Abelian group Subring

Definition

Let $R$ be a commutative ring. A grading of $R$ is a family $(R_n)_{n \ge 0}$ of subgroups of the abelian group $(R, +)$ such that:

$R = \bigoplus_{n \ge 0} R_n$ as abelian groups, that is, every $f \in R$ is uniquely a finite sum $f = \sum_n f_n$ with $f_n \in R_n$ ;
$R_m R_n \subseteq R_{m+n}$ for all $m, n \ge 0$ .

A graded ring is a ring with a grading. The elements of $R_n$ are the homogeneous elements of degree $n$ , and $f_n$ is the homogeneous component of $f$ of degree $n$ .

Remarks

Example: $R = k[x_1, \dots, x_r]$ , with $k$ a field and $R_n$ the span of the monomials of degree $n$ ; the product of two monomials of degrees $m$ and $n$ has degree $m + n$ . Non-example: in $R = k[x]/(x^2 - 1)$ take $R_0 = k$ , $R_1 = k \bar x$ and $R_n = 0$ for $n \ge 2$ ; clause 1 holds, but $\bar x \cdot \bar x = 1$ is a nonzero element of $R_0$ , not of $R_2 = 0$ , so clause 2 fails. It follows from the definition that $1 \in R_0$ and that $R_0$ is a subring. One also grades by $\Z$ or by another monoid in the same way; the ring $R$ is then $\Z$ -graded, and so on.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Abelian group concept
Commutative ring concept
Subring concept
Graded ring definition

Where this leads

1 page builds on this one.

Cobordism ring

Tag	`CAPQBQDRA7`
URI	`m2://CAPQBQDRA7`
Kind	definition
Writing	W0 stub
MSC	13A02
Defines	Graded ring
Uses	Commutative ring, Abelian group, Subring
Changed	2 November 2026

Graded ring

Definition

Remarks

Reading path

Where this leads

Further reading

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