Enveloping algebras that are principal ideal rings

TL;DR

This paper characterizes when the restricted enveloping algebra of a restricted Lie algebra over a field of positive characteristic is a principal ideal ring, linking it to the algebra's structure as an extension of a torus by a cyclic algebra.

Contribution

It provides a complete characterization of restricted enveloping algebras that are principal ideal rings in terms of the structure of the underlying Lie algebra.

Findings

Restricted enveloping algebra is a principal ideal ring iff the Lie algebra is an extension of a finite-dimensional torus by a cyclic algebra.

The result characterizes the algebraic structure necessary for the enveloping algebra to have principal ideals.

The paper establishes a structural criterion connecting Lie algebra extensions to ring-theoretic properties.

Abstract

Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.