In this paper we are dealing with associative triple systems of second kind in Loos' notation. We develop a general theory of the sacle 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) where X,Y are nomlegenerate inner product spaces overa unital associative algebra with involution (δT), where x,y are hermitian and either (1)δ=δ1is a division algebra or (2) δ=δ1⨁δ1opwith the exchange involution,or(3) X,Y are alternate δ = K isafield with the identity as involution. Finally we classify semi prime associative triple systems with descending chain condition on left (right) ideals. In particular, the structure theorem of Lms for semiprime associative triple systems with descendingchain condition on all inner ideels a n be derived from our results.