  Let D be a division algebra over a base field k. The homological
transcendence degree of D, denoted by Htr D, is defined to be the injective
dimension of the enveloping algebra of D. We show that Htr has several useful
properties which the classical transcendence degree has. We extend some results
of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several
classes of division algebras. The main tool for the computation is Van den
Bergh's rigid dualizing complex.
