Fitting decomposition in Jordan systems

Abstract

In this note, the well known Fitting decomposition is extended to the context of Jordan algebras, pairs, and triple systems. Recall Fitting’s Lemma: Letfbe an endomorphism of a module M such that the sequences f(M) 3f2(M) 1 . . . and Kerfc Ker f2 c . . . become stationary. Then M= M1 @M,, where M, =f”(M) and MO = Ker f” for all sufficiently large n. Now let c be the idempotent of A = End(M) given by projection of M onto M,, and let A = A,, 0 A,, 0 AOi 0 &,, be the corresponding Peirce decomposition of A, where aeAii if and only if a(Mj) cM, and u(M,-~)=O. Then f=fil +fooeA,, @A,, and fil =f IM, is invertible in Ali, whereas &,, = f 1 M,, is nilpotent. In this form, Fitting’s Lemma makes sense for Jordan algebras; in fact, if we consider A as a Jordan algebra J then the Jordan Peirce spaces of c are J, = A,,, J1 = A,, @ A,,, Jo = A,,, and invertibility in J2 and nilpotence in J are Jordan concepts. We prove (Theorem 1): If J is a Jordan algebra with dcc on principal inner ideals then for every a E J there exists a unique idempotent c of J such that a =a2 + a,,~ J2 @ Jo in the Peirce decomposition of J with respect to c, where a2 is invertible in J, and a, is nilpotent. In associative ring theory, Fitting’s Lemma implies that the endomorphism ring of an indecomposable module is local. The Jordan analogue is: A unital Jordan algebra with dcc on principal inner ideals containing no idempotents #O, 1, and such that this remains true in all isotopes, is local. The condition on the isotopes (which cannot be dropped, cf. [S]) indicates that this is really a result on Jordan pairs. Thus it is natural to ask for a Fitting decomposition in Jordan pairs which indeed exists just as in the algebra case, the sole difference being that the idempotent defining the Peirce decomposition is merely unique up to association (Theorem 2). Finally, there is a Fitting decomposition in Jordan triple