Modules over a complete discrete valuation ring

Abstract

1. Introduction. In this paper we prove a structure theorem for reduced countably generated R-modules of finite rank, where R is a complete discrete valuation ring (e.g., the p-adic integers). In addition, we obtain an existence theorem for such modules of rank 1. These two theorems are applied to cancellation, direct summand, and unique factorization problems. Our results are generalizations of theorems of Kaplansky-Mackey [2] and Rotman [3]. We observe that classification of modules up to isomorphism is not appropriate for mixed modules. Instead, we use a slightly weaker relation, almost isomorphism (which we define below). For example, although it is not true that a direct summand of a completely decomposable module is again completely decomposable, it is often true that the summand is almost isomorphic to a completely decomposable module. 2. Prerequisites. We give a summary of basic definitions in this section. A detailed account may be found in [1]. R is a discrete valuation ring (DVR) if it is a local principal ideal domain. R becomes a topological ring by defining the neighborhoods of 0 to be the powers of the prime ideal (p); R is complete if it is complete as a metric space. An arbitrary sequence in R either contains a convergent subsequence or contains a subsequence whose terms are of the form unpk, where k is a fixed non