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Ad-nilpotent Elements of Semiprime Rings with Involution
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 2 / 01 June 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 318-327
- Print publication:
- 01 June 2018
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- Article
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Let
$R$ be an 
$n!$-torsion free semiprime ring with involution 
$*$ and with extended centroid 
$C$, where 
$n\,>\,1$ is a positive integer. We characterize 
$a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying 
${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on 
$K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if 
$a,\,b\,\in \,R$ satisfy 
${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either 
$n$ is even or 
$b\,=\,0$, then 
${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some 
$\lambda \,\in \,C$.
