Local PI Theory of Jordan Systems II

We pursue the study, initiated in a previous paper, of Jordan systems having nonzero local algebras that satisfy a polynomial identity. We define the extended centroid of a nondegenerate Jordan system, the corresponding central extension, which we call the extended central closure, and prove a Jordan analogue of Martindaleʹs theorem on prime algebras having a generalized identity: If J is a nondegenerate Jordan system with nonzero PI-elements, then the extended central closure of J has nonzero socle, equal to its PI ideal.