2026-04-26T20:21:42Zhttps://uvadoc.uva.es/oai/request

oai:uvadoc.uva.es:10324/804852025-12-11T20:00:30Zcom_10324_1129com_10324_931com_10324_894col_10324_1193

Brox, Jose García, Esther Gómez Lozano, Miguel Muñoz Alcázar, Rubén Vera de Salas, Guillermo 2025-12-11T03:58:11Z 2025-12-11T03:58:11Z 2021 Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45, 631-646 0126-6705 https://uvadoc.uva.es/handle/10324/80485 10.1007/s40840-021-01206-8 631 2 646 Bulletin of the Malaysian Mathematical Sciences Society 45 2180-4206 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 $n$. spa info:eu-repo/semantics/openAccess Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution info:eu-repo/semantics/article