**Theorem**. Let be a -algebra and suppose that is a regular submonoid of contained in the center of Then

*Proof*. Let be a finitely generated -subalgebra of and suppose that is a frame of $T.$ Choose and such that for all Let be the -subalgebra of generated by and let be the -subspace generated by and Now, since is in the center of we have Thus Therefore

for every finitely generated -subalgebra of of and so On the other hand, because is regular, and thus

Using the above result we can now evaluate the GK dimension of a Laurent polynomial ring.

**Corollary**. Let be a -algebra and let be a variable over Then

*Proof*. Since is the localization of at the central regular submonoid we have The result now follows from Theorem 1.

Let be a -algebra. We showed that if and only if is locally finite. We also saw that there is no algebra of GK dimension strictly between 0 and one. Now, what can we say about the case ? There is a partial answer to this question and we will see it in the next post.