Graded ring: Difference between revisions

Latest revision as of 18:41, 3 January 2009

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

@@ Line 5: / Line 5: @@
 A '''graded ring''' is a [[commutative unital ring]] <math>A</math> equipped with a direct sum decomposition as a sum of Abelian subgroups:
-<math>A = \oplus_{i=0}^\infty A_i = A_0 \oplus A_1 \oplus A_2 \oplus \ldots</math>
+<math>A = \oplus_{i=-\infty}^\infty A_i = \cdots A_{-2} \oplus A_{-1} \oplus A_0 \oplus A_1 \oplus A_2 \oplus \cdots</math>
 such that the following hold:
@@ Line 13: / Line 13: @@
 * <math>A_mA_n \subset A_{m+n}</math>. In other words, if <math>a \in A_m</math> and <math>b \in A_n</math> then <math>ab \in A_{m+n}</math>
-A structure of the above sort on a ring is termed a '''gradation''', also a <math>\mathbb{Z}</math>-gradation.
+A structure of the above sort on a ring is termed a '''gradation''', also a <math>\mathbb{Z}</math>-gradation. The ring <math>A</math> is '''positively graded''' if <math>A_i = 0</math> for all <math>i<0</math>.
 There are related notions for noncommutative rings.