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 $Q_m^s(R)$. There exists an idempotent $e\in Q_m^s(R)$ that orthogonally decomposes $a=ea+(1-e)a$ and either both $ea$ 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 $\mathbb{Z}$-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 $n$.