Ad-nilpotent Elements of Semiprime Rings with Involution

Lee, Tsiu-Kwen

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 318-327

Résumé

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 $\lambda \,\in \,C$ .

Détail

DOI : 10.4153/CMB-2017-005-3

Mots-clés : 16N60, 16W10, 17B60, semiprime ring, Lie algebra, Jordan algebra, faithful, f-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan element