Idealizers in the Second Weyl Algebra

Publication date: 23 September 2020

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. In this paper we consider idealizers in the second Weyl algebra

A_{2}

, which is the ring of differential operators on

m a t h b b 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

f A_{2}

in

A_{2}

is always left and right noetherian, extending the work of McCaffrey.

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