Commutators and products of Lie ideals of prime rings
Tsiu-Kwen Lee, Jheng-Huei Lin
TL;DR
The paper extends Lie-ideal theory from simple to prime rings by establishing a Herstein-type dichotomy for simple rings and a suite of centralizer-based criteria in prime rings. It shows that in a simple ring a Lie ideal $L$ must satisfy $L\subseteq Z(R)$, $L=Z(R)a+Z(R)$ with $a\notin Z(R)$, or $[R,R]\subseteq L$, and introduces the notion of exceptional prime rings via the extended centroid $C$. For prime rings, the authors completely characterize when $L+aL$ or products of noncentral Lie ideals contain a nonzero ideal, and they relate vanishing commutators to centralizer conditions such as $KC=LC=Ca+C$ in exceptional cases. The results culminate in a set of theorems (B–E) that unify and extend existing Lie-ideal theory through centralizers, extended centroids, and exceptional- prime phenomena, offering new criteria for nontrivial ideals arising from Lie-ideal constructions.
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.