The Jordan-Zassenhaus Theorem states that, if R is a Dedekind domain whose field Q of quotients is a global field, then for each R-order S in a semisimple algebra over Q, and for each positive integer n, there are only finitely many isomorphism classes of left S-lattices of rank ≤n. This result (which plays an important role in the theory of orders and group representations) was used by Lady [L] (independently and simultaneously by C.M. Bang, unpublished) to show that a torsion-free abelian group of finite rank can have but a finite number of non-isomorphic direct summands. As Lady himself observed, his proof extends without change to torsion-free modules of finite rank over Dedekind domains R for which the Jordan-Zassenhaus theorem is valid.
Our main objective here is to find a precise relation between the Jordan-Zassenhaus theorem and the finiteness of direct decompositions of torsionfree modules of finite rank. We will show (see our Theorem 5.2) that for a Dedekind domain R of characteristic 0, the Jordan-Zassenhaus theorem holds if and only if finite rank torsion-free R-modules have, up to isomorphism, only finitely many direct summands and the factor rings R/I are 1 This paper was written while the second author was visiting the Department of Mathematics of Tulane University. He is grateful for their hospitality.