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 ${{\,\mathrm{Skew}\,}}(R,*)$ 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 ${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$ 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 ea and $(1-e)a$ are ad-nilpotent of the same index (in this case the index of ad-nilpotence of a in ${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$ 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 ${{\,\mathrm{Skew}\,}}(Q_m^s(R),*)$ 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.