# Dictionary Definition

graded adj : arranged in a sequence of grades or
ranks; "stratified areas of the distribution" [syn: ranked, stratified]

# User Contributed Dictionary

## English

### Verb

graded- past of grade

# Extensive Definition

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).

## Graded rings

A graded ring A is a ring
that has a direct sum
decomposition into (abelian) additive groups

- A = \bigoplus_A_n = A_0 \oplus A_1 \oplus A_2 \oplus \cdots

- A_s \times A_r \rightarrow A_.

- x \in A_s, y \in A_r \implies xy \in A_

- A_s A_r \subseteq A_.

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

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

- A/I = \bigoplus_(A_n + I)/I .

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

- M = \bigoplus_M_i ,

and

- A_iM_j \subseteq M_

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-
- E=\bigoplus_i E_i

- 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.

Graded algebras are much used in commutative
algebra and algebraic
geometry, homological
algebra and algebraic
topology. One example is the close relationship between
homogeneous polynomials and projective
varieties.

## G-graded rings and algebras

We can generalize the definition of a graded ring
using any monoid G as an
index set. A G-graded ring A is a ring with a direct sum
decomposition

- A = \bigoplus_A_i

- A_i A_j \subseteq A_

Remarks:

- A graded algebra is then the same thing as a N-graded algebra, where N is the monoid of non-negative integers.
- If we do not require that the ring have an identity element, semigroups may replace monoids.
- G-graded modules and algebras are defined in the same fashion as above.

Examples:

- A group naturally grades the corresponding group ring; similarly, monoid rings are graded by the corresponding monoid.
- A superalgebra is another term for a Z2-graded algebra. Examples include Clifford algebras. Here the homogeneous elements are either of degree 0 (even) or 1 (odd).

In category
theory, a G-graded algebra A is an object in the
category of G-graded vector spaces, together with a morphism
\nabla:A\otimes A\rightarrow Aof the degree of the identity of
G.

## Anticommutativity

Some graded rings (or algebras) are endowed with an anticommutative structure. This notion requires the use of a semiring to supply the gradation rather than a monoid. Specifically, a signed semiring consists of a pair (Γ, ε) where Γ is a semiring and ε : Γ → Z/2Z is a homomorphism of additive monoids. An anticommutative Γ-graded ring is a ring A graded with respect to the additive structure on Γ such that:- xy=(-1)ε (deg x) ε (deg y)yx, for all homogeneous elements x and y.

### Examples

- An exterior algebra is an example of an anticommutative algebra, graded with respect to the structure (Z≥ 0, ε) where ε is the homomorphism given by ε(even) = 0, ε(odd) = 1.
- A supercommutative algebra (sometimes called a skew-commutative associative ring) is the same thing as an anticommutative (Z/2Z, ε) -graded algebra, where ε is the identity endomorphism for the additive structure.

## See also

## References

- Algebra .

graded in German: Graduierung (Algebra)

graded in Spanish: Álgebra graduada

graded in French: Algèbre graduée

graded in Hebrew: אלגברה מדורגת

graded in Russian: Градуированная
алгебра

# Synonyms, Antonyms and Related Words

aligned, arranged, arrayed, assorted, cataloged, categorized, classified, composed, constituted, disposed, filed, fixed, grouped, harmonized, hierarchic, indexed, marshaled, methodized, normalized, on file, ordered, orderly, organized, pigeonholed, placed, pyramidal, ranged, ranked, rated, regularized, regulated, routinized, sorted, standardized, stratified, synchronized, systematized, tabular