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

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=ea+(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 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 .