Jacobson radical non-nil algebras of Gel’fand-Kirillov dimension 2

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 Gel’fand-Kirillov dimension two. This refutes a conjecture incorrectly attributed to Goodearl.