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

• Published in: Bulletin of the Malaysian Mathematical Sciences Society Vol. 44; no. 4; pp. 2577 - 2602
• Main Authors: Brox, Jose, García, Esther, Lozano, Miguel Gómez, Alcázar, Rubén Muñoz, de Salas, Guillermo Vera
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01-07-2021; Springer Nature B.V
• Subjects: Applications of Mathematics; Associativity; Lie groups; Mathematics; Mathematics and Statistics; Rings (mathematics); Semiprime ring; 17B60; Skew-symmetric elements; Lie algebra; 16W10; Ad-nilpotent element; 16N60
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-020-01064-w

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
∈
R
is a pure ad-nilpotent element of
R
of index
n
with
R
free of
t
and
n
t
-torsion for
t
=
[
n
+
1
2
]
, then
n
is odd and there exists
λ
∈
C
(
R
)
such that
a
-
λ
is nilpotent of index
t
. If
R
is a semiprime ring with involution
∗
and
a
is a pure ad-nilpotent element of
Skew
(
R
,
∗
)
free of
t
and
n
t
-torsion for
t
=
[
n
+
1
2
]
, then either
a
is an ad-nilpotent element of
R
of the same index
n
(this may occur if
n
≡
1
,
3
(
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
≡
0
,
3
(
mod
4
)
). The case
n
≡
2
(
mod
4
)
is not possible.