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# Ad-nilpotent elements of semiprime rings with involution

Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$.
 Keywords: Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan element