Maximal commutative subrings and simplicity of Ore extensions

TL;DR

This paper characterizes when Ore extension rings, especially differential polynomial rings, are simple, linking their simplicity to properties of their centers and the base rings, with new conditions for maximal commutative subrings.

Contribution

It provides necessary and sufficient conditions for the simplicity of Ore extensions, focusing on differential polynomial rings and the role of maximal commutative subrings.

Findings

Differential polynomial ring is simple iff its center is a field and R is -simple.

The centralizer of R in R[x;,elta] is a maximal commutative subring containing R.

If R is -simple and maximal commutative, then R[x;,elta] is simple.

Abstract

The aim of this article is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x;id,\delta], is simple if and only if its center is a field and R is \delta-simple. When R is commutative we note that the centralizer of R in R[x;\sigma,\delta] is a maximal commutative subring containing R and, in the case when \sigma=id, we show that it intersects every non-zero ideal of R[x;id,\delta] non-trivially. Using this we show that if R is \delta-simple and maximal commutative in R[x;id,\delta], then R[x;id,\delta] is simple. We also show that under some conditions on R the converse holds.