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graded ring (Definition)

Let $ S$ be a groupoid (semigroup,group) and let $ R$ 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 $ R$ with $ s \in S$. If $ R_{s} R_{t} \subseteq R_{st}$ for all $ s,t \in S$ then we say that $ R$ is groupoid graded (semigroup-graded, group-graded) ring.

We refer to $ R = \bigoplus_{s\in S} R_{s}$ as an $ S$-grading of $ R$ and the subgroups $ R_{s}$ as the $ s$-components of $ R$. 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 $ R$ is strongly graded by $ S$.

Any element $ r_{s}$ in $ R_{s}$ (where $ s\in S$) is said to be homogeneous of degree $ s$. 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 $ G$ is a subsemigroup of $ S$ then $ R_{G}$ is a subring of $ R$. If $ G$ is a left (right, two-sided) ideal of $ S$ then $ R_{G}$ is a left (right, two-sided) ideal of $ R$.

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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See Also: homogeneous ideal, support (graded ring)

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 8 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 8369 times total.

Classification:
AMS MSC: 13A02 (Commutative rings and algebras :: General commutative ring theory :: Graded rings)
Pending Errata and Addenda
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