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# Ad-nilpotent Elements of Semiprime Rings with Involution

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Published:2017-01-30
Printed: Jun 2018
• Tsiu-Kwen Lee,
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
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## Abstract

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
 MSC Classifications: 16N60 - Prime and semiprime rings [See also 16D60, 16U10] 16W10 - Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx] 17B60 - Lie (super)algebras associated with other structures (associative, Jordan, etc.) [See also 16W10, 17C40, 17C50]

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