Affine rings of low GK dimension

We consider algebras of low GK dimension. We give a new, completely combinatorial proof that a finitely generated domain of GK dimension 1 must be a finite module over its center (Theorem 2.4.2). We also show that the monic localization of a polynomial ring over a left Noetherian ring is a Jacobson ring (Theorem 2.3.28). We show that any subfield of the quotient ring of a finitely graded non-PI Goldie algebra of GK dimension 2 over a field F must have transcendence degree at most 1 over F (Theorem 3.3.19). In the fourth chapter we give counter-examples to several questions in ring theory. We construct a prime affine algebra of GK dimension 2 that is neither primitive nor PI; we construct a prime affine algebra of GK dimension 3 that has non-nil Jacobson radical; we construct a primitive affine algebra of GK dimension 3 with center that is not a field; and for each positive integer m we construct an affine algebra of GK dimension 2 with classical Krull dimension equal to m.