LAUSR

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 $$a\in R$$ a ∈ R is a pure ad-nilpotent element of R of index n with R free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ n t -torsion for $$t=[\frac{n+1}{2}]$$ t = [ n + 1 2 ] , then n is odd and there exists $$\lambda \in C(R)$$ λ ∈ C ( R ) such that $$a-\lambda $$ a - λ is nilpotent of index t . If R is a semiprime ring with involution $$*$$ ∗ and a is a pure ad-nilpotent element of $${{\,\mathrm{Skew}\,}}(R,*)$$ Skew ( R , ∗ ) free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ n t -torsion for $$t=[\frac{n+1}{2}]$$ t = [ n + 1 2 ] , then either a is an ad-nilpotent element of R of the same index n (this may occur if $$n\equiv 1,3 \,(\text {mod } 4)$$ n ≡ 1 , 3 ( mod 4 ) ) or R is a nilpotent element of R of index $$t+1$$ t + 1 , and R satisfies a nontrivial GPI (this may occur if $$n\equiv 0,3 \,(\text {mod } 4)$$ n ≡ 0 , 3 ( mod 4 ) ). The case $$n\equiv 2 \,(\text {mod } 4)$$ n ≡ 2 ( mod 4 ) is not possible.