# Graded algebra

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 **grading**.

## Graded algebra

A **graded algebra** *A* is an algebra that has a direct sum decomposition

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \bigoplus_{n\in \mathbb N}A_i = A_0 \oplus A_1 \oplus A_2 \oplus \cdots}**

such that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_m A_n \subseteq A_{m + n}.}**

Elements of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n}**
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.

Since rings may be regarded as **Z**-algebras, a **graded ring** is defined to be a graded **Z**-algebra.

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*,*T*^{n}*V*. - 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*H*^{n}.

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 algebra

We can generalize the definition of a graded algebra to an arbitrary monoid *G* as an index set. A *G*-graded algebra*A* is an algebra with a direct sum decomposition

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such that

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i A_j \subseteq A_{i \cdot j} }**

A graded algebra is then the same thing as a **N**-graded algebra, where **N** is the monoid of natural numbers.

(If we don't require that the ring has an identity element, we can extend the definition from monoids to semigroups.

Examples of G-graded algebras include:

- The group ring of a group is naturally graded by that group; similarly, monoid rings are graded by the corresponding monoid.
- A superalgebra is another term for a
**Z**_{2}-graded algebra. Clifford algebras are a common family of examples. Here the homogeneous elements are either even (degree 0) or odd (degree 1).

Category theoretically, a *G*-graded algebra *A* is an object in the category of *G*-graded vector spaces together with a morphism **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla:A\otimes A\rightarrow A}**
of the degree of the identity of *G*.

## Graded modules

The corresponding idea in module theory is that of a **graded module**, namely a module *M* over *A* such that also

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and

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This idea is much used in commutative algebra, and elsewhere, to define under mild hypotheses a **Hilbert function**, namely the length of *M*_{n} 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).

## See also

de:Graduierung (Algebra) es:Álgebra graduada ru:Градуированная алгебра