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Brox-López, José Ramón García, Esther Gómez-Lozano, Miguel Ángel Muñoz-Alcázar, Rubén José Vera de Salas, Guillermo 2024-09-25T17:04:04Z 2024-09-25T17:04:04Z 2022-03 Brox, J., García, E., Gómez Lozano, M. et al. Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution. Bull. Malays. Math. Sci. Soc. 45, 631–646 (2022). https://doi.org/10.1007/s40840-021-01206-8 https://hdl.handle.net/10630/33313 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 W-m(s) (R). There exists an idempotent e is an element of Q(m)(s) (R) that orthogonally decomposes a = ea + (1 - e)a and either ea and (1 - e)a are ad-nilpotent of the same index (in this case the index of adnilpotence 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. eng open access Álgebra Ad-Nilpotent Elements of Skew Index in Semiprime Rings with Involution journal article