Jordan centroids

Central simple triples are important for the classification of prime Jordan triples of Clifford type in arbitrary, characterstics. For triples and pairs (or even for unital Jordan algebras of characteristic 2), there is no workable notion of center, and the concept of "central simple" system must be understood as "centroid-simple", The centroid of a Jordan system (algebra, triple, or pair) consists of the "natural" scalars for that system: the largest unital, commutative ring Gamma such that the system can be considered as a quadratic Jordan system over Gamma. In this paper we will characterize the centroids of the basic simple Jordan algebras, triples, and pairs. (Consideration of the tangled ample outer ideals in Jordan algebras of quadratic forms will be left to a separate paper.) A powerful tool is the Eigenvalue Lemma, that a centroidal transformation on a prime system over Phi which has an eigenvalue alpha in Phi must actually be scalar multiplication by alpha. An important consequence is that a prime system over Phi with reduced elements P(x)J = Phi x (or which grows reduced elements under controlled conditions) must already be central, Gamma = Phi.