As usual, I’ll assume that is a field. Recall that if a -algebra is an Ore domain, then we can localize at and get the division algebra The algebra is called the quotient division algebra of

**Theorem (**Borho and Kraft**, **1976) Let be a finitely generated -algebra which is a domain of finite GK dimension. Let be a -subalgebra of and suppose that Let Then is an Ore subset of and Also, is finite dimensional as a (left or right) vector space over

*Proof*. First note that, by the corollary in this post, is an Ore domain and hence both and exist and they are division algebras. Now, suppose, to the contrary, that is not (left) Ore. Then there exist and such that This implies that the sum is direct for any integer Let be a frame of a finitely generated subalgebra of Let and suppose that is the subalgebra of which is generated by For any positive integer we have

and thus because the sum is direct. So and hence Taking supremum of both sides over all finitely generated subalgebras of will give us the contradiction A similar argument shows that is right Ore. So we have proved that is an Ore subset of Before we show that we will prove that is finite dimensional as a (left) vector space over So let be a frame of For any positive ineteger let Clearly for all and

because So we have two possibilities: either for some or the sequence is strictly increasing. If then we are done because is finite dimensional over and hence is finite dimensional over Now suppose that the sequence is strictly increasing. Then because Fix an integer and let be a -basis for Clearly we may assume that for all Let be a frame of a finitely generated subalgebra of Then

which gives us

because the sum is direct. Therefore which is a contradiction. So we have proved that the second possibility is in fact impossible and hence is finite dimensional over Finally, since, as we just proved, the domain is algebraic over and thus it is a division algebra. Hence because and is the smallest division algebra containing