graded ring

Let S be a groupoid (semigroup,group) and let R be a ring (not necessarily with unity) which can be expressed as a R=sSRs of additive subgroups Rs of R with sS. If RsRtRst for all s,tS then we say that R is groupoid graded (semigroup-graded, group-graded) ring.

We refer to R=sSRs as an S-grading of R and the subgroups Rs as the s-components of R. If we have the stronger condition that RsRt=Rst for all s,tS, then we say that the ring R is strongly graded by S.

Any element rs in Rs (where sS) is said to be homogeneous of degree s. Each element rR can be expressed as a unique and finite sum r=sSrs of homogeneous elementsPlanetmathPlanetmathPlanetmath rsRs.

For any subset GS we have RG=gGRg. Similarly rG=gGrg. If G is a subsemigroup of S then RG is a subring of R. If G is a left (right, two-sided) ideal of S then RG 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 ringsMathworldPlanetmath
filtered algebras

Title graded ring
Canonical name GradedRing
Date of creation 2013-03-22 11:45:03
Last modified on 2013-03-22 11:45:03
Owner aplant (12431)
Last modified by aplant (12431)
Numerical id 19
Author aplant (12431)
Entry type Definition
Classification msc 13A02
Classification msc 16W30
Classification msc 14L15
Classification msc 14L05
Classification msc 12F10
Classification msc 11S31
Classification msc 11S15
Classification msc 11R33
Synonym S-graded ring
Synonym G-graded ring
Related topic HomogeneousIdeal
Related topic SupportGradedRing
Defines groupoid graded ring
Defines semigroup graded ring
Defines group graded ring
Defines homogeneous element
Defines strongly graded