Examples in finite Gelʹfand–Kirillov dimension

By modifying constructions of Be dar and Small we prove that for countably generated prime F-algebras of finite GK dimension there exists an affinization having finite GK dimension. Using this result we show: for any field there exists a prime affine algebra of GK dimension two that is neither primitive nor PI; for any countable field F there exists a prime affine F-algebra of GK dimension three that has non-nil Jacobson radical; for any countable field F there exists an affine primitive F-algebra of GK dimension at most four with center equal to a polynomial ring; for a countable field F there exists a primitive affine Jacobson F-algebra of GK dimension three that does not satisfy the Nullstellensatz.