Herstein's theorems and simplicity of hermitian Jordan systems

In this paper we extend Herstein's first construction relating associative and Jordan ideals to pairs and triple systems. As a consequence we show tha t an associative pair or triple system is simple if and only if its Jordan symmetrization is simple. We also generalize Herstein's second construction to ample subsystems of associative algebras, pairs, and triple systems, wh ich provides information on their simplicity when the associative structure is simple. (C) 2001 Elsevier Science.