- Graded algebra
In

mathematics , in particularabstract algebra , a**graded algebra**is analgebra over a field (orcommutative ring ) with an extra piece of structure, known as a**gradation**(or "grading").**Graded rings**A

**graded ring**"A" is a ring that has adirect sum decomposition into (abelian) additive groups:$A\; =\; igoplus\_\{nin\; mathbb\; N\}A\_n\; =\; A\_0\; oplus\; A\_1\; oplus\; A\_2\; oplus\; cdots$such that the ring multiplication maps:$A\_s\; imes\; A\_r\; ightarrow\; A\_\{s\; +\; r\}.$Explicitly this means that:$x\; in\; A\_s,\; y\; in\; A\_r\; implies\; xy\; in\; A\_\{s+r\}$and so:$A\_s\; A\_r\; subseteq\; A\_\{s\; +\; r\}.$Elements of $A\_n$ 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:$A/I\; =\; igoplus\_\{nin\; mathbb\; N\}(A\_n\; +\; I)/I$.

Any (non-graded) ring "A" can be given a gradation by letting "A"

_{0}= "A", and "A"_{"i"}= 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:$M\; =\; igoplus\_\{iin\; mathbb\; N\}M\_i\; ,$

and

:$A\_iM\_j\; subseteq\; M\_\{i+j\}$

This idea is much used in

commutative algebra , and elsewhere, to define under mild hypotheses a, namely the length of "M"Hilbert function _{"n"}as a function of "n". Again under mild hypotheses of finiteness, this function is a polynomial, theHilbert polynomial , for all large enough values of "n" (see alsoHilbert-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::$E=igoplus\_i\; E\_i$such that

#"A"_{i}"E"_{j}⊂ "E"_{i+j}, and

#"E"_{i}"E"_{j}⊂ "E"_{i+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 ring s. The homogeneous elements of degree "n" are exactly the homogeneouspolynomial s of degree "n".

* Thetensor algebra "T"^{•}"V" of avector space "V". The homogeneous elements of degree "n" are the tensors of rank "n", "T"^{"n"}"V".

*Theexterior algebra Λ^{•}"V" andsymmetric algebra "S"^{•}"V" are also graded algebras.

* Thecohomology ring "H"^{•}in anycohomology theory is also graded, being the direct sum of the "H"^{"n"}.Graded algebras are much used in

commutative algebra andalgebraic geometry ,homological algebra andalgebraic topology . One example is the close relationship between homogeneouspolynomial s 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:$A\; =\; igoplus\_\{iin\; G\}A\_i$such that:$A\_i\; A\_j\; subseteq\; A\_\{i\; cdot\; j\}$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,semigroup s may replacemonoid s.

*"G"-graded modules and algebras are defined in the same fashion as above.Examples:

* A group naturally grades the correspondinggroup ring ; similarly,monoid ring s are graded by the corresponding monoid.

*Asuperalgebra is another term for a**Z**_{2}-graded algebra. Examples includeClifford algebra s. 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 $abla:Aotimes\; A\; ightarrow\; A$of 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 asemiring to supply the gradation rather than a monoid. Specifically, a**signed semiring**consists of a pair (Γ, ε) where Γ is asemiring and ε : Γ →**Z**/2**Z**is ahomomorphism 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.

*Asupercommutative algebra (sometimes called a**skew-commutative associative ring**) is the same thing as an anticommutative (**Z**/2**Z**, ε) -graded algebra, where ε is the identity endomorphism for the additive structure.**See also***

graded vector space

*graded category

*differential graded algebra

*graded Lie algebra

*filtered algebra , a generalization**References***.

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