l Introduction, A p-primary abelian group is a module over the p-adic integers; thus Ulm's theorem can be viewed as a classification of reduced countably generated torsion modules over the p-adic integers, or, more generally, over a complete discrete valuation ring. It is with this point of view that Kaplansky and Mackey [4] generalized Ulm's theorem to cover mixed modules of rank 1. In this paper their result is generalized in various ways, sometimes to modules of finite rank, sometimes to

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... les over possibly incomplete rings. The structure theorems obtained are applied to solve square-root, cancellation, and direct summand problems. The main idea is to squeeze as much information as possible from the proof of Ulm's theorem in [4] . In order to understand our procedure, we sketch that proof. Order, once for all, generating sets of the modules T and T f : The plan is to build an isomorphism stepwise up these lists. The crucial point is then, given a height-preserving isomorphism f:S->S',S finitely generated, to extend / to a height-preserving isomorphism of {t l9 S} and a suitable submodule of T" containing S'. In order to construct this extension it is necessary to normalize ί t in two ways: (i) assume pt t e S; (ii) assume that ί 4 has maximal height in the coset t t + S. If T is torsion, both of these normalizations are always possible. Now the" possibility of extra generality arises precisely at these two points. If T is mixed and (ii) is satisfied, then the proof will go through if T/S is torsion; this is what Kaplansky and Mackey did in their paper. In this paper, we define a class of modules in which (ii) can always be satisfied, and it is this class of modules which we shall consider. 2 Definitions, A discrete valuation ring (DVR) is a principal ideal domain R with a unique prime ideal (p). Γ\n=i(P n ) = (0). Hence if r e R is non-zero, there is a maximal n, depending on r, such that r e (p n ). Define \r\ = e~n; define |0| = 0. | | is a norm which satisfies the strong triangle inequality: \r + r'\ < max \r\, \r'\. This norm induces a metric on R. R is a complete DVR if it is complete in this metric. If R is incomplete, we may form its completion i2*, and JS* is a complete DVR. The p-adic integers is a complete DVR; it is also compact as a metric space. Let Q be the quotient field of R. We define the rank of a module M (often called the 'torsion-free rank') to be the dimension of the Q