 Almost commutative algebra

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.