On Lie nilpotent rings and Cohen's Theorem

TL;DR

This paper investigates Lie nilpotent rings, establishing properties of their radicals and ideals, and extends Cohen's theorem to this context, also exploring Lie centers and their generated subrings.

Contribution

It introduces a Lie nilpotent version of Cohen's theorem and analyzes properties of radicals, ideals, and Lie centers in Lie nilpotent rings.

Findings

Prime radical equals the set of nilpotent elements.

Left ideals plus the prime radical form two-sided ideals.

A Lie nilpotent version of Cohen's theorem is established.

Abstract

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z_n(R), which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring of R generated by the union of Z_n(R) and C is also Lie nilpotent of index n.