Modules of Finite Rank Over Prufer Rings

Abstract

In this paper the results of [4] are rederived and completed (in particular the question raised on p. 338 is completely settled) by a systematic use of extension theory [2, 3]. Instead of handling Dedekind rings and valuation rings separately, the treatment throughout is in terms of Prilfer rings; the modified descending chain condition in the one case and the hypothesis of almost maximality in the other are reformulated as linear precompactness (which serves also to exhibit their relationship to each other). The technique for obtaining the decomposition theorems is by and large the same: search for a pure submodule and proof that it is a direct summand. However, the concept of purity undergoes a closer analysis which permits one to disengage the relevant hypotheses. The rather successful definition of rank might also be mentioned. We shall be dealing exclusively with modules over commutative integral domains R for which the unit is the identity endomorphism. The most general rings we shall consider are the Prufer rings in which every finitely generated ideal is invertible: among these are the finitely principal domains (every two elements have a greatest common divisor) and,' more particularly, the valuation rings (in which this divisor is equal to one of the two elements). A module M is torsion-free if for every non-zero x E M, ax = 0 implies a = 0. A torsion-free module can be embedded in a vector space over Q, the quotient field of R; the dimension of the smallest subspace containing M (this is independent of the embedding) is called the rank of M. The rank of an arbitrary module M is defined as the smallest of the ranks of the torsion-free modules of which M is a quotient module. A module M is divisible if for every x E M and a E R (a $ 0) there exists y E M such that ay = x. This concept is in a way dual to that of torsion-free; in a divisible module the non-zero scalar multiplications are onto endomorphisms while in a torsion-free module they are one-to-one. The field Q (and more generally any vector space), as well as its quotient modules, are divisible; thus the rank of a module could also be defined as the smallest rank of a divisible module containing it.