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 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 L in the extended centroid such that a-L 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=1,3 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=0,3 mod 4). The case is n=2 mod 4 not possible.