Transcendence Degree of Division Algebras

2010/02/26 by Bell, Jason P.
#16P90 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1002.4915

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.

Related

• Gelfand-Kirillov dimension of commutative subalgebras of simple infinite dimensional algebras and their quotient division algebras
• Division algebras of Gelfand-Kirillov dimension two
• Homological Transcendence Degree
• The Transcendence Degree over a Ring
• Gelfand-Kirillov dimension for rings