Local-to-global inheritance of primitivity in Jordan algebras

Abstract

We show that a strongly prime Jordan algebra having a primitive local algebra is primitive. As a tool, similar results concerning one-sided primitivity and Ã-primitivity of associative algebras are established.Introduction. The notion of local algebra of a Jordan system, introduced in [17], has proved to be a basic tool in the study of primitivity [3,4,7]. Unlike the algebra case, primitivity of a Jordan pair is considered at a particular element, and shown to be equivalent to primitivity of its local algebra at this element, under the assumption of strong primeness [4]. As a consequence, the results on local inheritance of primitivity in Jordan algebras [9] were an important tool to have some possibility of changing the element at which primitivity holds, which is essential in the description of primitive Jordan pairs and triple systems given in [4].In this paper we show the inheritance of primitivity by a strongly prime Jordan algebra from any primitive local algebra. Apart from its intrinsic interest, this result will play an important role in the definite solution of the above problem on the ªlocationº of primitivity for Jordan systems [5], as one could expect.The main ingredients of the proofs are the general properties of primitivity and its local inheritance in Jordan algebras [8,9], as well as some facts concerning primitive pairs and triple systems [3,4]. After a preliminary section, collecting known results which will be used in the sequel, we begin in Section 1 with the study of local-to-global inheritance of one-sided primitivity and Ã-primitivity in associative algebras. The corresponding result for Jordan algebras splits into two cases: the hermitian case and the ªfiniteº case. Due to the fact that the initial object is the local algebra, the above two cases are those of non-homotope-PI and homotope-PI Jordan algebras. This forces us to use the theory of pairs and triple systems, so that results on the relationship of a Jordan algebra and its underlying triple system are needed. Non-homotope-PI Jordan algebras are studied in Section 2 using the main result of Section 1, while the homotope-PI case in Section 3 follows directly by an argument involving Peirce decompositions and the socle. We unify the different cases in the main theorem of Section 4, getting also an important consequence on inheritance of primitivity from Peirce components.

Local and Subquotient Inheritance of Simplicity in Jordan Systems

“…5 The previous result yields a characterization of simple regular pairs in terms of the simplicity of their subquotients:…”

“…associative triple system. 5. Local algebras of Jordan and associative systems, introduced in 21 , are one of the most useful tools to study primitivity of Jordan pairs and triple systems.…”

“…verses of 2.1 , 2.5 , 2.6 , and 3. 5 . Though our purpose is studying Jordan systems, we have included in this section, for the sake of completeness, associative versions of the Jordan results.…”

“…Now Q V y s Q V y rKer M s Q qS y s Q qI y : I q .We have shown that every nonzero ideal of S contains the nonzero ideal Ž Q V , S , i.e., H s Q V , S . On the other hand, for any b g S , S ( V is simple by 2 5. .…”