Decomposability of modules

Abstract

It is an old theorem, essentially due to Prufer [5 ], that any torsionfree module of countable rank over a complete discrete valuation ring is a direct sum of modules of rank one. In [2] I generalized this to maximal valuation rings. In seeking to perfect this theorem, it is natural to ask: for what integral domains is it true that any torsionfree module of rank two is a direct sum of modules of rank one? There does not appear to be a simple answer. By changing the problem slightly, and confining the investigation to Noetherian domains, it is possible to single out the pleasant class of rings of the theorem below. However, the real reason for publishing this note is that the proof of Theorem 19 in [3 ] is, to put it politely, unconvincing.