Theorem 1. Let be a -algebra. If is a variable over then
Proof. Let be a finitely generated subalgebra of generated by Let be the subalgebra of generated by the coefficients of Then clearly is a finitely generated subalgebra of and Now, let be a frame of Let Then is a generating subspace of and clearly for all integers Hence and so
Therefore It is also clear that for all integers Thus and so
Therefore and the result follows.
Corollary 1. Let be a -algebra. Then
An immediate result of corollary 2 is that if is an infinite set of commuting variables, then
So, by corollary 2, for any integer there exists an algebra such that In fact, for any real number there exists a -algebra such that It is also known that if then there is no algebra with This result is called the Bergman’s gap theorem. We now look at the GK dimension of noncommutative polynomial algebras.
Theorem 2. If is a set of noncommuting variables with then
Proof. Let and consider the -subalgebra of Choose the generating subspace Then it follows easily that and thus
Corollary. Let be a -algebra which is a domain. If then is Ore.
Proof. If contains a copy of where and are noncommuting variables, then by Theorem 2. Thus does not contain such a subalgebra and hence is an Ore domain by the lemma in this post.