Transcendence Degree of Division Algebras

Details Transcendence Degree of Division Algebras Transcendence Degree of Division Algebras

2010-02-26

arxiv.org/abs/1002.4915v1

Mathematics

Rings and Algebras

10 pages

Scientific paper

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 $D$.

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