PlanetMath: graded ring

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graded ring

(Definition)

Let be a groupoid (semigroup ,group) and let be a ring (not necessarily with unity) which can be expressed as a direct sum $R = {\bigoplus}_{s \in S} R_{s}$ of additive subgroups $R_{s}$ of with $s \in S$ . If $R_{s} R_{t} \subseteq R_{st}$ for all $s,t \in S$ then we say that is groupoid graded (semigroup-graded, group-graded) ring.

We refer to $R = \bigoplus_{s\in S} R_{s}$ as an -grading of and the subgroups $R_{s}$ as the -components of . If we have the stronger condition that $R_{s}R_{t} = R_{st}$ for all $s,t \in S$ , then we say that the ring is strongly graded by .

Any element $r_{s}$ in $R_{s}$ (where $s\in S$ ) is said to be homogeneous of degree . Each element $r \in R$ can be expressed as a unique and finite sum $r = \sum_{s \in S} r_{s}$ of homogeneous elements $r_{s} \in R_{s}$ .

For any subset $G \subseteq S$ we have $R_{G} = \sum_{g \in G} R_{g}$ . Similarly $r_{G} = \sum_{g \in G} r_{g}$ . If is a subsemigroup of then $R_{G}$ is a subring of . If is a left (right, two-sided) ideal of then $R_{G}$ is a left (right, two-sided) ideal of .

Some examples of graded rings include:
Polynomial rings
Ring of symmetric functions
Generalised matrix rings
Morita contexts
Ring of Hirota derivatives
group rings
filtered algebras

"graded ring" is owned by aplant. [ full author list (2) | owner history (1) ]

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Other names:

S-graded ring, G-graded ring

Also defines:

groupoid graded ring, semigroup graded ring, group graded ring, homogeneous element, strongly graded

Keywords:

algebra ring groupoid homogeneous

Attachments:: homogeneous ideal (Definition) by archibal
support (graded ring) (Definition) by aplant

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Cross-references: filtered algebras, group rings, derivatives, matrix rings, functions, symmetric, polynomial rings, ideal, right, subring, subsemigroup, subset, sum, finite, homogeneous of degree, subgroups, additive, unity, ring, group, semigroup, groupoid
There are 10 references to this entry.

This is version 14 of graded ring, born on 2001-10-15, modified 2007-09-07.
Object id is 192, canonical name is GradedRing.
Accessed 7994 times total.

Classification:

AMS MSC:

13A02 (Commutative rings and algebras :: General commutative ring theory :: Graded rings)

Pending Errata and Addenda

None.[ View all 9 ]

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