Nondegenerate Jordan Algebras Satisfying Local Goldie Conditions

Abstract

A structure theorem is given for nondegenerate Jordan algebrasJsatisfying the ascending chain condition on annihilators of a single element and such thatJcontains no infinite direct sum of inner deals inside the inner ideal generated by each elementx∈J. As a consequence of this theorem and of the main results of a previous paper by the authors (J. Algebra174(1995), 1024–1048), it is obtained that such Jordan algebrasJare precisely the local orders in nondegenerate Jordan algebras satisfying dcc on principal inner ideals and without non-artinian quadratic ideals, which extends to local orders the Zel'manov theorem for Goldie Jordan algebras.