Prime and Primitive Algebras with Prescribed Growth Types

TL;DR

This paper extends previous work on monomial algebras by constructing prime and primitive finitely generated algebras with various prescribed growth types, including examples with non-zero locally nilpotent ideals.

Contribution

It demonstrates how to modify existing constructions to produce prime algebras with arbitrary growth and introduces methods to build primitive algebras with regular growth types.

Findings

Constructed prime algebras with prescribed growth types.

Developed a method to produce primitive algebras with regular growth.

Created examples of prime algebras containing non-zero locally nilpotent ideals.

Abstract

Bartholdi and Smoktunowicz constructed finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type. Moreover, using a construction of an inverse system of monomial ideals which arise from this construction we are able to further construct finitely generated primitive algebras with many regular enough growth types. Then, inspired by an example by Zelmanov, we show how our prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of the primitive constructions.