Commutators and products of Lie ideals of prime rings

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 $Z\left(R\right)$, then $L\subseteq Z\left(R\right)$, $L=Z\left(R\right)a+Z\left(R\right)$ 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 $K_1,\dots ,K_m$ with $m\ge 2$ such that $K_1{K}_2\cdots K_m$ contains a nonzero ideal of $R$. Finally, we characterize noncentral Lie ideals $K_j$’s and $L_k$’s satisfying $\left[K_1{K}_2\cdots K_m,L_1{L}_2\cdots L_n\right]=0$ from the viewpoint of centralizers.