Graded algebra

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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).

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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 satisfies

and so

Elements of An are known as homogeneous elements of degree n. An ideal or other subset \mathfrak{a}A is homogeneous if for every element a\mathfrak{a}, the homogeneous parts of a are also contained in \mathfrak{a}.

If I is a homogeneous ideal in A, then A / I is also a graded ring, and has decomposition

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

An algebra A over a ring R is a graded algebra if it is graded as a ring. In the case where the ring R is also a graded ring, then one requires that

Note that the definition of the graded ring over a ring with no grading is the special case of the latter definition where "R" is given the trivial grading (every element of "R" is of grade 0).

Examples of graded algebras are common in mathematics:

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