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

Abstract

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.