17 - Transcendental Division Algebras

Summary

In this chapter we study another very “classical” topic, namely, transcendental division algebras (that is, division rings which are not algebraic over their centers). While at first glance it may not appear that the general theory of noetherian rings has anything to say about division rings, we shall see that much concrete information can be gained by applying noetherian methods to polynomial rings over division rings, in particular by applying what we have learned in previous chapters about injective modules, Ore localizations, and Krull dimension. We shall, for instance, derive analogs of the Hilbert Nullstellensatz for polynomial rings over division rings and over fully bounded rings. Information about a division ring D with center k will then be obtained by developing connections among the transcendence degree of D over k, the question of primitivity of a polynomial ring D[x₁, …, x_n], and the Krull dimension of D ⊗_k k(x₁, …, x_n), as well as connections between the noetherian condition on D ⊗_k D and the question of finite generation of subfields of D. For technical reasons, and in order to be able to apply some of these results to Goldie quotient rings, we actually derive most of the results in this chapter for simple artinian rings rather than for division rings.