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Graded
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). more...
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Graded rings
A graded ring A is a ring that has a direct sum decomposition into (abelian) additive groups
 such that the ring multiplication maps
- Explicitly this means that whenever
- and so
Elements of An are known as homogeneous elements of degree n. An ideal, or other set in A, is homogeneous if for every element a it contains, the homogeneous parts of a are also contained in it.
Any (non-graded) ring A can be given a gradation by letting A0 = A, and Ai = 0 for i > 0. This is called the trivial gradation on A.
Graded modules
The corresponding idea in module theory is that of a graded module, namely a module M over a graded ring A such that also
- and
- This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a Hilbert function, namely the length of Mn as a function of n. Again under mild hypotheses of finiteness, this function is a polynomial, the Hilbert polynomial, for all large enough values of n (see also Hilbert-Samuel polynomial).
Graded algebras
A graded algebra over a graded ring A is an A-algebra E which is both a graded A-module and a graded ring in its own right. Thus E admits a direct sum decomposition
-
 such that
AiEj ⊂ Ei+j, and;
EiEj ⊂ Ei+j.;
Often when no grading on A is specified, it is assumed that A receives the trivial gradation, in which case one may still talk about graded algebras over A without risk of confusion.
Examples of graded algebras are common in mathematics:
Polynomial rings. The homogeneous elements of degree n are exactly the homogeneous polynomials of degree n.;
The tensor algebra T•V of a vector space V. The homogeneous elements of degree n are the tensors of rank n, TnV.;
The exterior algebra Λ•V and symmetric algebra S•V are also graded algebras.;
The cohomology ring H• in any cohomology theory is also graded, being the direct sum of the Hn.;
Read more at Wikipedia.org
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