Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution

• Published in: Bulletin of the Malaysian Mathematical Sciences Society Vol. 45; no. 2; pp. 631 - 646
• Main Authors: Brox, Jose, García, Esther, Gómez Lozano, Miguel, Muñoz Alcázar, Rubén, Vera de Salas, Guillermo
• Format: Journal Article
• Language: English
• Published: Singapore Springer Singapore 01.03.2022; Springer Nature B.V
• Subjects: Applications of Mathematics; Centroids; Mathematics; Mathematics and Statistics; Quotients; Rings (mathematics); GPI; Involution; 16R50; Semiprime ring; 16W25; Grading; Matrix ring; 16W10; Ad-nilpotent element
• ISSN: 0126-6705; 2180-4206
• DOI: 10.1007/s40840-021-01206-8

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 ∈ Q m s ( R ) that orthogonally decomposes a = e a + ( 1 - e ) a and either 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 Z -grading induced by a ∗ -complete family of orthogonal idempotents and that e Q 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 .