Graded ring - Revision history

Masnevets: /* Definition */ added notion of positively graded

2009-01-03T18:41:05Z

Definition: added notion of positively graded

← Older revision		Revision as of 18:41, 3 January 2009
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	A '''graded ring''' is a [[commutative unital ring]] <math>A</math> equipped with a direct sum decomposition as a sum of Abelian subgroups:		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:		such that the following hold:
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	* <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>		* <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.		There are related notions for noncommutative rings.

Vipul: 2 revisions

2008-05-12T16:22:50Z

2 revisions

← Older revision	Revision as of 16:22, 12 May 2008
(No difference)

Vipul at 21:53, 8 February 2008

2008-02-08T21:53:31Z

← Older revision		Revision as of 21:53, 8 February 2008
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			{{ring with addl structure}}

	==Definition==		==Definition==

Vipul: New page: ==Definition== A '''graded ring''' is a commutative unital ring $A$ equipped with a direct sum decomposition as a sum of Abelian subgroups: $A = \oplus_{i=0}^\infty A...$

2008-02-08T21:07:47Z

New page: ==Definition== 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...

New page

==Definition==

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>

such that the following hold:

* Each <math>A_i</math> is a subgroup under addition
* <math>1 \in A_0</math>
* <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.

There are related notions for noncommutative rings.

==Related notions==

===Weaker notions===

* [[Filtered ring]]

Graded ring - Revision history

Masnevets: /* Definition */ added notion of positively graded

Vipul: 2 revisions

Vipul at 21:53, 8 February 2008

Vipul: New page: ==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=0}^\infty A...

Vipul: New page: ==Definition== A '''graded ring''' is a commutative unital ring $A$ equipped with a direct sum decomposition as a sum of Abelian subgroups: $A = \oplus_{i=0}^\infty A...$