PRIME ASSOCIATIVE TRIPLE-SYSTEMS WITH NONZERO SOCLE

In this paper we are dealing with associative triple systems of second kind in Loos’ notation. We develop a general theory of the socle paralleling that for associative and Jordan algebras and give a structure theorem for semiprime associative triple systems whose socle is an essential ideal. In particular we prove that every prime associative triple system A containing minimal inner ideals is isomorphic to a triple of continuous linear operators containing all finite rank operators, L(X,Y)⊃A⊃f(X,Y) whereX, Y are nondegenerate inner product spaces over a unital associative algebra with involution (Δ,τ), where X, Y are hermitian and either (1) Δ = Δ1is a division algebra or (2) Δ=Δ1⊕ Δ1op with the exchange involution, or (3) X,Y are alternate and Δ=K is a field with the identity as involution. Finally we classify semiprime associative triple systems with descending chain condition on left (right) ideals. In particular, the structure theorem of Loos for semiprime associative triple systems with descending chain condition on all inner ideals can be derived from our results. © 1990, Taylor & Francis Group, LLC. All rights reserved.