Centralizers in Ore extensions of polynomial rings

TL;DR

This paper investigates the structure of centralizers in Ore extensions of polynomial rings, proving they are commutative, finitely generated, and sometimes singly generated, with implications for algebraic structure understanding.

Contribution

It establishes that centralizers in Ore extensions are commutative and finitely generated, and identifies conditions for them to be singly generated.

Findings

Centralizers are commutative.

Centralizers are finitely generated.

Some centralizers are singly generated.

Abstract

In this paper we consider centralizers of single elements in Ore extensions of the ring of polynomials in one variable over a field. We show that they are commutative and finitely generated as an algebra. We also show that for certain classes of elements their centralizer is singly generated as an algebra.