Primary decomposable subspaces of $k[t]$ and Right ideals of the first Weyl algebra $A_{1}(k)$ in characteristic zero

TL;DR

This paper characterizes right ideals of the first Weyl algebra over any characteristic zero field using the concept of primary decomposable subspaces of polynomial rings, extending previous results to more general fields.

Contribution

It introduces the notion of primary decomposable subspaces of $k[t]$ to describe right ideals of the Weyl algebra over arbitrary fields of characteristic zero, generalizing prior work.

Findings

Description of right ideals of $A_1$ over any characteristic zero field.

Introduction of primary decomposable subspaces of $k[t]$.

Generalization of Cannings and Holland's results to non-algebraically closed fields.

Abstract

In this article, we describe the right ideals of $A_1:=k[t,\partial]$, the first Weyl agebra, over any field $k$ of characteristic zero. For this, we define the notion of primary decomposable subspaces of $k[t]$. This description generalizes a result of Cannings and Holland obtained for an algebraically closed field $k$. Dans cet article, on d\'ecrit les id\'eaux \`a droite de $A_1$ sur un corps quelconque de caract\'eristique nulle. Pour cela on d\'efinit la notion de sous-espaces d\'ecomposables primaires de $k[t]$. Cette description g\'en\'eralise un r\'esultat de Cannings et Holland obtenu pour un corps $k$ alg\'ebriquement clos.