Ore extensions of commutative rings and the Dixmier-Moeglin equivalence

TL;DR

This paper investigates the structure of Ore extensions over commutative rings, establishing conditions under which the Dixmier-Moeglin equivalence holds, especially for Gelfand-Kirillov dimension less than four, and provides counterexamples for higher dimensions.

Contribution

It proves the Dixmier-Moeglin equivalence for Ore extensions with Gelfand-Kirillov dimension under four and identifies cases where it fails for higher dimensions.

Findings

Dixmier-Moeglin equivalence holds for dimension less than four.

Counterexamples exist for dimension four or higher.

Prime ideals are primitive iff they are locally closed and have certain center properties.

Abstract

We consider Ore extensions of the form $T:=R[x;\sigma,\delta]$ with $R$ a commutative integral domain that is finitely generated over a field $k$. We show that if $T$ has Gelfand-Kirillov dimension less than four then a prime ideal $P\in {\rm Spec}(T)$ is primitive if and only if $\{P\}$ is locally closed in ${\rm Spec}(T)$, if and only if the Goldie ring of quotients of $T/P$ has centre that is an algebraic extension of $k$. We also show that there are examples for which these equivalences do not all hold for $T$ of integer Gelfand-Kirillov dimension greater than or equal to $4$.