On a conjecture of Goodearl: Jacobson radical non-nil algebras of   Gelfand-Kirillov dimension 2
  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. 
Agata Smoktunowicz, Laurent Bartholdi
math.RA