A Description of Ad-nilpotent Elements in Semiprime Rings with Involution

Autor: Esther García, Miguel Gómez Lozano, Jose Brox, Rubén Muñoz Alcázar, Guillermo Vera de Salas
Rok vydání: 2021
Předmět:
Zdroj: Bulletin of the Malaysian Mathematical Sciences Society. 44:2577-2602
ISSN: 2180-4206
0126-6705
DOI: 10.1007/s40840-020-01064-w
Popis: In this paper, we study ad-nilpotent elements in Lie algebras arising from semiprime associative rings R free of 2-torsion. With the idea of keeping under control the torsion of R, we introduce a more restrictive notion of ad-nilpotent element, pure ad-nilpotent element, which is a only technical condition since every ad-nilpotent element can be expressed as an orthogonal sum of pure ad-nilpotent elements of decreasing indices. This allows us to be more precise when setting the torsion inside the ring R in order to describe its ad-nilpotent elements. If R is a semiprime ring and $$a\in R$$ is a pure ad-nilpotent element of R of index n with R free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ -torsion for $$t=[\frac{n+1}{2}]$$ , then n is odd and there exists $$\lambda \in C(R)$$ such that $$a-\lambda $$ is nilpotent of index t. If R is a semiprime ring with involution $$*$$ and a is a pure ad-nilpotent element of $${{\,\mathrm{Skew}\,}}(R,*)$$ free of t and $$\left( {\begin{array}{c}n\\ t\end{array}}\right) $$ -torsion for $$t=[\frac{n+1}{2}]$$ , then either a is an ad-nilpotent element of R of the same index n (this may occur if $$n\equiv 1,3 \,(\text {mod } 4)$$ ) or R is a nilpotent element of R of index $$t+1$$ , and R satisfies a nontrivial GPI (this may occur if $$n\equiv 0,3 \,(\text {mod } 4)$$ ). The case $$n\equiv 2 \,(\text {mod } 4)$$ is not possible.
Databáze: OpenAIRE
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