RT info:eu-repo/semantics/article
T1 Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution
A1 Brox, Jose
A1 García, Esther
A1 Gómez Lozano, Miguel
A1 Muñoz Alcázar, Rubén
A1 Vera de Salas, Guillermo
AB In this paper we study ad-nilpotent elements of semiprime rings $R$ with involution $*$ whose indices of ad-nilpotence differ on $\Skew(R,*)$ and $R$. The existence of such an ad-nilpotent element $a$ implies the existence of a GPI of  $R$, and determines a big part of its structure. When moving to the symmetric Martindale ring of quotients $Q_m^s(R)$ of $R$,$a$ remains ad-nilpotent of the original indices in $\Skew(Q_m^s(R),*)$ and $Q_m^s(R)$. There exists an idempotent $e\in Q_m^s(R)$ that orthogonally decomposes $a=ea+(1-e)a$ and either both $ea$ and $(1-e)a$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of $a$ in $\Skew(Q_m^s(R),*)$ is congruent with 0 modulo 4), or $ea$ and $(1-e)a$ have different indices of  ad-nilpotence (in this case the index of ad-nilpotence of $a$ in $\Skew(Q_m^s(R),*)$ is congruent with 3 modulo 4). Furthermore we show that  $Q_m^s(R)$ has a finite $\mathbb{Z}$-grading induced by a $*$-complete family of orthogonal idempotents and that $eQ_m^s(R)e$, which contains $ea$, is isomorphic to a ring of matrices over its extended centroid. All this information is used to produce examples of these types of ad-nilpotent elements for any possible index of ad-nilpotence $n$.
PB Springer Nature
SN 0126-6705
YR 2021
FD 2021
LK https://uvadoc.uva.es/handle/10324/80485
UL https://uvadoc.uva.es/handle/10324/80485
LA spa
NO Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45, 631-646
DS UVaDOC
RD 11-ene-2026