Quasi-duo differential polynomial rings

TL;DR

This paper characterizes when differential polynomial rings are quasi-duo, showing they are symmetric in left and right cases, providing examples, and describing their maximal ideals, addressing a question from 2005.

Contribution

It offers a complete characterization of quasi-duo differential polynomial rings and clarifies their structure, including the non-existence in multiple indeterminates.

Findings

A differential polynomial ring is left quasi-duo iff it is right quasi-duo.

Complete description of maximal ideals in quasi-duo differential polynomial rings.

No quasi-duo differential polynomial rings exist with multiple indeterminates.

Abstract

In this article we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide non-trivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.