Abstract view
Adnilpotent Elements of Semiprime Rings with Involution


Published:20170130
Printed: Jun 2018
TsiuKwen Lee,
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
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, adnilpotent element, Jordan element
Semiprime ring, Lie algebra, Jordan algebra, faithful $f$free, involution, skew (symmetric) element, adnilpotent element, Jordan element

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