Jacobson radical algebras with Gelfand–Kirillov dimension two over countable fields

It is shown that for every countable field K, there is a finitely generated graded Jacobson radical algebra over K of Gelfand–Kirillov dimension two. Examples of finitely generated Jacobson radical algebras of Gelfand–Kirillov dimension two over algebraic extensions of finite fields of characteristic 2 were earlier constructed by Bartholdi [L. Bartholdi, Branch Rings, thinned rings, tree enveloping rings, Israel J. Math. (in press)].