Transcendence Degree of Division Algebras

TL;DR

This paper introduces a new notion of transcendence degree for division algebras, extending Zhang's concept, and uses it to prove a conjecture relating Gelfand-Kirillov dimensions of subalgebras.

Contribution

It defines a noncommutative transcendence degree for division algebras and applies it to prove a conjecture about Gelfand-Kirillov dimensions in Ore domains.

Findings

Defined a new invariant with desirable properties.

Proved the conjecture of Small regarding Gelfand-Kirillov dimensions.

Established a noncommutative analogue of transcendence degree.

Abstract

We define a transcendence degree for division algebras, by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let $k$ be a field, let $A$ be a finitely generated $k$-algebra that is an Ore domain, and let $D$ denote the quotient division algebra of $A$. If $A$ does not satisfy a polynomial identity then the Gelfand-Kirillov dimension of $K$ is at most the Gelfand-Kirillov dimension of $A$ minus 1 for every commutative subalgebra $K$ of $D$.