Prime ideals in differential operator rings and crossed products of infinite groups

Abstract

We use a version of the Martindale ring of quotients to study prime ideals in extensions of a ringR corresponding to two cases. IfG is a group of automorphisms ofR we form the crossed product R ∗ G. If g is a Lie algebra of derivations of R we have the twisted differential operator ring, denoted by R ∗ g (sometimes known as the “twisted” smash product R# t u ( g )). We obtain analogues of Incomparability for crossed products of nilpotent groups, and differential operator rings of solvable Lie algebras. In the case of crossed products, the incomparability result of D. S. Passman and M. Lorenz for the infinite cyclic group (also proved by G. Bergman) is generalized.