On a conjecture of Goodearl: Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2

TL;DR

The paper constructs a specific algebra over a countable field that disproves Goodearl's conjecture by exhibiting a prime, non-nil Jacobson radical algebra with Gelfand-Kirillov dimension two.

Contribution

It provides a counterexample to Goodearl's conjecture, showing such algebras can exist with the specified properties.

Findings

Constructed a graded algebra over a countable field with the given properties.

Demonstrated the algebra is Jacobson radical but not nil.

Refuted the conjecture by Goodearl.

Abstract

For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl.