Almost commutative algebra ( $gr A is commutative and finitely generated$ )

An almost commutative algebra over a field is an algebra with an exhaustive increasing nonnegative multiplicative filtration whose associated graded ring is commutative and finitely generated as an algebra over the field. A common presentation uses finitely many degree-one generators, scalar degree-zero part, and commutators that lower degree. The finite-generation requirement is essential: an infinitely generated commutative polynomial ring has commutative associated graded algebra but need not be Noetherian.

Table of contents
- Degree-one almost commutative algebra Almost commutative algebra

Degree-one almost commutative algebra ( $F_{0} A = k, A = ⟨ F_{1} A ⟩, [F_{1} A, F_{1} A] \subseteq F_{1} A$ )

 0  0

This is the degree-one convention for an almost commutative algebra: the exhaustive filtration is generated by a finite-dimensional first step containing

1

, the degree-zero part is

k

, and its associated graded algebra is commutative. The first step is a finite-dimensional Lie algebra under commutators. Its inclusion into

A

extends by the universal property to a surjection from its universal enveloping algebra. The element

1

of the first step is retained as a central Lie generator and its image is the associative identity; quotienting the first step by

k 1

before taking the bracket can lose scalar commutators. This convention is stronger than merely requiring a finitely generated commutative associated graded algebra for an arbitrary weighted filtration.

// Destination: noncommutative-algebra.bigb