Graded ring: Difference between revisions

Revision as of 21:53, 8 February 2008

This article defines a notion of a ring with additional structure

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

$A = \oplus_{i = 0}^{\infty} A_{i} = A_{0} \oplus A_{1} \oplus A_{2} \oplus \dots$

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 $a b \in A_{m + n}$

A structure of the above sort on a ring is termed a gradation, also a $Z$ -gradation.

There are related notions for noncommutative rings.

Revision as of 21:07, 8 February 2008 (view source) Vipul (talk \| contribs) (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...)		Revision as of 21:53, 8 February 2008 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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			{{ring with addl structure}}

	==Definition==		==Definition==