Strict proper nilness modulo an absorber

An element z of a Jordan system J is strictly properly nilpotent if it is nilpotent in all homotopes of all extensions of J; it is strictly properly nilpotent of bounded index if there is a bound on its indices of nilpotence in all these extensions. We showed in a previous paper that the strictly properly nilpotent elements (resp. those of bounded index) form an ideal, by exhibiting that ideal as an Amitsur shrinkage. In this paper we prove by combinatorial methods (the exponential law for power series and Jordan Binomial Theorems) a modular generalization of this: the elements strictly properly nilpotent (resp. boundedly so) modulo the absorber Q of an inner ideal K form an ideal. As a corollary we obtain Zelmanov’s Nilness-mod-the-Absorber-Theorem that the ideal generated by Q is nil mod K. From this we re-derive his Primitive Absorber and Exceptional Heart Theorems (improved to show the heart is S(J), not just S( (J)³), two results crucial to the structure of primitive Jordan systems, using a triple-system version of Zelmanov's KKT-specialization for inner ideals in a Jordan pair.