For the definition of a PI-algebra see this post.

**Lemma**. Let be a field and let be a finitely generated -algebra which is a domain. If is not PI, then

*Proof*. If then is finite dimensional over and so over its center, and hence it is PI. If then is again PI (see here). Also, there is no algebra whose GK dimension is strictly between 1 and 2, by the Bergman’s gap theorem. Thus

We are now going to refine the result given in the above lemma. We will write for the GK dimension of an algebra viewed as an -algebra if is not

**Theorem**. (Smith and Zhang, 1996) Let be a field and let be a finitely generated -algebra which is a domain. If is not PI, then where is the center of

*Proof*. By the above lemma, we may assume that and Let be the central localization of and let be the center of Clearly is just the quotient field of Recall, from the theorem in here, that

and

Let and Then there exist a finite dimensional -vector subspace of which contains 1 and for all large enough integers Also, there exists a finite dimensional -vector subspace of which contains 1 and for all large enough integers Hence, for large enough integers we have

Thus Since the above inequality holds for all real number and we have

Now, since is finitely generated as a -algebra, is a finitely generated -algebra and clearly is not PI because is not PI. Therefore by the above lemma. We also have and now the result follows from