Primitive Jordan Pairs and Triple Systems

Abstract

In this paper we give a characterization of primitivity of Jordan pairs and triple systems in terms of their local algebras. As a consequence of that local characterization we extend to Jordan pairs and triple systems most of the known results about primitive Jordan algebras. In particular, we describe primitive Jordan pairs and triple systems over an arbitrary ring of scalars in the sense of ''The Structure of Primitive Quadratic Jordan Algebras'' by J. A. Anquela, T. Cortes, and Ž .

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 .…”

“…We will deal with Jordan pairs and triple systems over an arbitrary ring of scalars Φ. The reader is referred to [3,4,7,10] for basic results, notation and terminology, though we will stress some definitions and basic notions.…”

Local PI Theory of Jordan Systems II

“…nonzero socle in fact, they equal their socles , both options can be expressed by the elementary sentences 0. 3 « P P P P P P P¨/ 0 & P P P P P P P P P r / 0 Ž . nonzero elements a g PI J l I and set a s a q a .…”

“…3. Extended central closures of linear algebras can be seen in light of the notions of localization theory, through the category of modules over the corresponding Ž w x. multiplication algebras and some of its subcategories see 31 . In our case, however, multiplication algebras do not seem to be adequate tools for studying extended central closures since they do not reflect the nonlinearity of multiplications in Jordan systems.…”

Growing Hearts in Associative Systems

“…Associative pairs are defined accordingly. The reader is referred to [1,2,8,9,10,11] for basic facts and notions not explicitly mentioned in this section. 0.2 Given an associative system R, the heart Heart(R) of R is the intersection of all nonzero ideals of R. It can be readily seen that, when R is semiprime, Heart(R) is the unique simple ideal of R when it is nonzero [5, 2.1, 3.1, 3.7].…”