Localization and ideal theory in iterated differential operator rings

Abstract

In this paper we study primality, hypercentrality, simplicity, and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings. We give sufficient conditions for the skew enveloping algebra of a nilpotent Lie algebra with coefficient ring containing the rational numbers to be a simple ring, and we give necessary and sufficient conditions in the case that the Lie algebra is Abelian. Our main results show that if L is a finite dimensional solvable Lie algebra over a field k of characteristic zero and R is an Artinian or a commutative Noetherian algebra over k, then the skew enveloping algebra R# U( L) satisfies the second layer condition. We discuss consequences of this for localization and use the localization theory to state a classical Krull dimension versus global dimension inequality when k is uncountable.