Commutators and products of Lie ideals of prime rings

TL;DR

This paper investigates the structure of Lie ideals in prime and simple rings, providing characterizations of their properties, products, and commutators, thus generalizing classical results and deepening understanding of ring theory.

Contribution

It offers new characterizations of Lie ideals and their interactions in prime rings, extending classical theorems and exploring products and commutators of noncentral Lie ideals.

Findings

Lie ideal in a simple ring is either central, generated by a noncentral element, or contains the commutator subgroup

Characterization of Lie ideals whose sum with a scalar multiple contains a nonzero ideal

Conditions under which the commutator of two noncentral Lie ideals is zero

Abstract

Motivated by some recent results on Lie ideals, it is proved that if $L$ is a Lie ideal of a simple ring $R$ with center $Z(R)$, then $L\subseteq Z(R)$, $L=Z(R)a+Z(R)$ for some noncentral $a\in L$, or $[R, R]\subseteq L$, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let $R$ be a prime ring with extended centroid $C$. We completely characterize Lie ideals $L$ and elements $a$ of $R$ such that $L+aL$ contains a nonzero ideal of $R$. Given noncentral Lie ideals $K, L$ of $R$, it is proved that $[K, L]=0$ if and only if $KC=LC=Ca+C$ for any noncentral element $a\in L$. As a consequence, we characterize noncentral Lie ideals $K_1,\ldots,K_m$ with $m\geq 2$ such that $K_1K_2\cdots K_m$ contains a nonzero ideal of $R$. Finally, we characterize noncentral Lie ideals $K_j$'s and $L_k$'s satisfying $\big[K_1K_2\cdots K_m, L_1L_2\cdots L_n\big]=0$ from the viewpoint of centralizers.