Graded ring

This article defines a notion of a ring with additional structure

Definition

A graded ring is a commutative unital ring $A$ equipped with a direct sum decomposition as a sum of Abelian subgroups:

$A=\oplus _{i=-\infty }^{\infty }A_{i}=\cdots A_{-2}\oplus A_{-1}\oplus A_{0}\oplus A_{1}\oplus A_{2}\oplus \cdots$

such that the following hold:

• Each $A_{i}$ is a subgroup under addition
• $1\in A_{0}$
• $A_{m}A_{n}\subset A_{m+n}$. In other words, if $a\in A_{m}$ and $b\in A_{n}$ then $ab\in A_{m+n}$

A structure of the above sort on a ring is termed a gradation, also a $\mathbb {Z}$-gradation. The ring $A$ is positively graded if $A_{i}=0$ for all $i<0$.

There are related notions for noncommutative rings.

Related notions

Weaker notions

• Filtered ring