Hersteinʹs Theorems and Simplicity of Hermitian Jordan Systems,

In this paper we extend Hersteinʹs first construction relating associative and Jordan ideals to pairs and triple systems. As a consequence we show that an associative pair or triple system is simple if and only if its Jordan symmetrization is simple. We also generalize Hersteinʹs second construction to ample subsystems of associative algebras, pairs, and triple systems, which provides information on their simplicity when the associative structure is simple.