Simple algebras of Gelfand-Kirillov dimension two

Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension $1$. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if $A$ is a finitely generated simple domain of quadratic growth, then $A$ is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed.