Ad-nilpotent Elements of Semiprime Rings with Involution. Issue 2 (1st June 2018)

• Record Type: Journal Article
• Authors: Lee, Tsiu-Kwen
• Is Part Of: Canadian mathematical bulletin =. Volume 61:Issue 2(2018)
• Page Start: 318
• Page End: 327
• Publication Date: 2018-06-01
• Subjects: 16N60 -- 16W10 -- 17B60; semiprime ring -- Lie algebra -- Jordan algebra -- faithful -- f-free -- involution -- skew (symmetric) element -- ad-nilpotent element -- Jordan element; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2017-005-3
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: 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 $\lambda \, \in \, C$ .