A result by Kulikov that does not extend to modules over general valuation domains

Abstract

Let R be a valuation domain. It is proved that every nonzero R-module contains a nonzero pure uniserial submodule if and only if R is rank-one discrete. It is a classical result in the theory of Abelian groups, due to Kulikov [5], that every Abelian p-group contains a pure cocyclic subgroup (which is necessarily a summand, cocyclic groups being pure-injective). Note that cocyclic p-groups are exactly the torsion uniserial Zp-modules, where 7p is the rank-one discrete valuation domain obtained by localizing the ring of the integers Z at the prime P. The goal of this paper is to show that this result is not extendable to valuation domains which are not rank-one discrete. Our main result is the following: Theorem. Let R be a valuation domain. Then every nonzero R-module contains a nonzero pure uniserial submodule if and only if R is rank-one discrete. The first contribution to the proof of this theorem was given in [2] (see also [3, X.4]), where an example of a cohesive R-module (i.e., a module without elements of limit height) with no nonzero uniserial pure submodules was given, under the assumption that the maximal ideal P of R is not principal. The second author showed in [6] that a cohesive R-module with the above property cannot exist if and only if J * RJ is a principal ideal of RJ, for each prime ideal J of R (equivalently, R is discrete and Spec(R) is well ordered by the opposite inclusion); note that the proof of the sufficiency was just an adaptation of the example given in [2]. The consequence of the above results is that, in order to complete the proof of the theorem, we must find a noncohesive R-module with no nonzero pure uniserial submodules, where R is any discrete valuation domain with a wellordered prime spectrum consisting of at least two nonzero ideals. In fact, we shall construct an R-module S, where R is as in the preceding Received by the editors October 16, 1989 and, in revised form, March 22, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 13C05, 13C12.