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 two 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. The Jacobson radical is very important for the study of noncommutative algebras. For a given ring R one usually studies the Jacobson radical J(R) of R, and the semiprimitive part R/J(R). As related evidence of a connection between these notions, a result of Amitsur says that the Jacobson radical of a finitely generated algebra over an uncountable field is nil, and it is known that all nil rings are Jacobson radical. It is important to know when Jacobson radical are nil because nil rings have interesting properties. For example subalgebras of nil algebras are nil, which does not hold in general for Jacobson radical rings. The Jacobson radical is important for determining the structure of rings and is a generalization of the Wedderburn radical for finitely dimensional algebras.