Noetherianity of idealizers in skew extensions

Abstract

Given a right ideal I in a ring R, the idealizer of I in R is the largest subring of R in which I becomes a two-sided ideal. These rings are of interest since they often give examples of rings with asymmetrical properties, and they may also display other pathological behaviour. For example, they provide many examples of noncommutative rings which are right but are not left noetherian. In this thesis we examine the behaviour of idealizers in skew extensions of commutative rings. We first focus on the skew group ring B = C#G, where C is a commutative noetherian domain and G is a finitely generated abelian group. For a prime ideal I of C, we study the idealizer of the right ideal IB in B and we obtain necessary and sufficient conditions for when the idealiser is left and right noetherian and we relate them to geometry, deriving interesting properties. We also give an example of these conditions in practice which translates to a curious number theoretic problem. We next consider idealizers in the second Weyl algebra A2, which is the ring of differential operators on k[x, y] (in characteristic 0). Specifically, let f be a polynomial in x and y which defines an irreducible curve whose singularities are all cusps. We show that the idealizer of the right ideal fA2 in A2 is always left and right noetherian, extending the work of McCaffrey.