A Description of Ad-nilpotent Elements in Semiprime Rings with Involution

Abstract

In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and $a\in R$ is a pure ad-nilpotent element of R of index n with R free of t and $\left( {\begin{array}{c}n\\ t\end{array}}\right)$-torsion for $t=[\frac{n+1}{2}]$, then n is odd and there exists $\lambda \in C(R)$ such that $a-\lambda$ is nilpotent of index t. If R is a semiprime ring with involution $*$ and a is a pure ad-nilpotent element of ${{\,\mathrm{Skew}\,}}(R,*)$ free of t and $\left( {\begin{array}{c}n\\ t\end{array}}\right)$-torsion for $t=[\frac{n+1}{2}]$, then either a is an ad-nilpotent element of R of the same index n (this may occur if $n\equiv 1,3 \,(\text {mod } 4)$) or R is a nilpotent element of R of index $t+1$, and R satisfies a nontrivial GPI (this may occur if $n\equiv 0,3 \,(\text {mod } 4)$). The case $n\equiv 2 \,(\text {mod } 4)$ is not possible.