graded algebra
A graded algebra is an algebra sorted into layers by 'degree', built so that multiplying respects the layers in a bookkeeping sense — multiply something of degree p by something of degree q and you always land in degree p + q. Polynomials graded by total degree are the everyday example: a quadratic times a cubic is always a quintic.
Formally, an N-graded algebra A over R is an R-algebra with a direct-sum decomposition A = ⊕_{d≥0} A_d into R-submodules (the homogeneous components of degree d) such that 1 ∈ A_0 and A_p · A_q ⊆ A_{p+q} for all p, q. Elements of a single A_d are called homogeneous of degree d. One may grade by Z, by an abelian group, or — for super-algebras — by Z/2Z; the additivity of degree under multiplication is the defining axiom in each case.
Grading is the organizing principle behind the central constructions of this subject: the tensor algebra T(V), exterior algebra Λ(V), and symmetric algebra S(V) are all graded by tensor degree, and a polynomial ring R[x_1, ..., x_n] is graded by total degree. Gradings also power Hilbert series, projective geometry (homogeneous coordinate rings), and the sign rule of graded-commutative algebras, where ab = (−1)^{pq} ba for homogeneous a, b of degrees p, q.
R[x, y] is graded by total degree: A_0 = R (constants), A_1 = span{x, y}, A_2 = span{x^2, xy, y^2}, with dim A_d = d + 1. The product x·y^2 of a degree-1 and a degree-2 element lands in degree 3, as the grading demands.
The polynomial ring is the prototypical graded algebra; degrees add under multiplication.
A graded ideal (homogeneous ideal) is one generated by homogeneous elements; quotient by such an ideal inherits the grading, which is precisely why Λ(V) and S(V) — quotients of T(V) by homogeneous ideals — are themselves graded. Quotienting by a non-homogeneous ideal generally destroys the grading.