Graded ring

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field, or more general R-algebra, in which there is a consistent notion of the weight of an element. The idea is that the weights of elements should add, when elements are multiplied. One has to allow the 'inconsistent' addition of elements of different weights, though. A formal definition follows.

Let G be an monoid. A G-graded algebra A is an algebra with a direct sum decomposition

<math>A = \bigoplus_{i\in G}A_i <math>

such that

<math> A_i A_j \subseteq A_{i \cdot j} <math>

An element of the ith subspace A_i is said to be a homogeneous (or pure) element of degree i.

Important examples of graded algebras include the tensor algebra T^•V of a vector space V as well as the exterior algebra Λ^•V both of which are N-graded. Given a commutative ring R, the polynomial ring R[x] over R is N-graded, being the direct sum of Rxⁿ for n ∈ N; further, any discrete valuation ring is graded by powers of the maximal ideal; further still, at least when the elements in the monoid G lack inverses, the monoid ring R[G] will be G-graded. The cohomology ring H^• in any cohomology theory is also graded by N, being the direct sum of the Hⁿ.

Clifford algebras and superalgebras are examples of Z₂-graded algebras. Here the homogeneous elements are either even (degree 0) or odd (degree 1).

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.

See also: graded vector space