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.

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

$\displaystyle A = \bigoplus_{n\in \mathbb N}A_i = A_0 \oplus A_1 \oplus A_2 \oplus \cdots$

such that

$\displaystyle A_m A_n \subseteq A_{m + n}.$

Elements of $\displaystyle A_n$ 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:

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.

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

$\displaystyle A = \bigoplus_{i\in G}A_i$

such that

$\displaystyle A_i A_j \subseteq A_{i \cdot j}$

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.

• 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 Z2-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 $\displaystyle \nabla:A\otimes A\rightarrow A$ of the degree of the identity of G.

$\displaystyle M = \bigoplus_{n\in \mathbb N}M_i ,$
$\displaystyle A_iM_j \subseteq M_{i+j}$