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 is an element of R is a pure ad-nilpotent element of R of index n with R free of t and ((n)(t))-torsion for t = [n+1/2], then n is odd and there exists lambda is an element of 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 Skew(R,*) free of t and ((n)(t))-torsion for t=[n+1/2], then either a is an ad-nilpotent element of R of the same index n (this may occur if n degrees 1,3(mod4)) or R is a nilpotent element of R of index t+1, and R satisfies a nontrivial GPI (this may occur if n degrees 0,3(mod4)). The case n degrees 2(mod4) is not possible