Local and Subquotient Inheritance of Simplicity in Jordan Systems

Abstract

In this paper we prove that the local algebras of a simple Jordan pair are simple. Jordan pairs all of which local algebras are simple are also studied, showing that they have a nonzero simple heart, which is described in terms of powers of the original pair. Similar results are given for Jordan triple systems and algebras. Finally, we characterize the inner ideals of a simple pair which determine simple Ž subquotients, answering the question posed by O. Loos and E. Neher 1994, J.. Algebra 166, 255᎐295 . ᮊ

Peirce gradings of Jordan systems

“…The proof follows from the corresponding results on inheritance of regularity conditions by subquotients [3,5,6,8], once it is shown that V 0 and V 2 are isomorphic to certain subquotients of V .…”

“…However, the proof given in this paper cannot be considered an alternative to the proof by McCrimmon in [9]. Indeed the results of this paper are based on the results of [5], which depend on the inheritance of regularity by local algebras proved in [6]. These latter results are based on the results by McCrimmon [9].…”

Imbedding Lie Algebras in Strongly Prime Algebras

“…This was aimed at building counterexamples linked to some problems with Jordan systems [1]. Recently, in [3], the results obtained in [2] were improved in two senses: on the one hand, associative systems over more general rings of scalars were considered, and, on the other hand, analogues for systems with involution were also obtained.…”

Growing Hearts in Associative Systems

“…Consequently, there is a long tradition of mathematical work aimed at linking Jordan notions with their corresponding associative ones. In [3], we studied how simple was a Jordan system having simple all of its local algebras. As a tool, it was proved that an associative system R with this condition had a big simple heart equal to the Jordan cube of R, i.e., (R (+) ) 3 .…”

“…In [3], we studied how simple was a Jordan system having simple all of its local algebras. As a tool, it was proved that an associative system R with this condition had a big simple heart equal to the Jordan cube of R, i.e., (R (+) ) 3 . Then in [4] we faced the problem of expressing Jordan cubes in terms of associative powers R n = R · · · R. Few new things could be said and the paper consisted, basically, on giving counterexamples to every (apparently) reasonable statement which came to mind.…”