Uniserial modules over valuation rings

Abstract

This is the first in a series of papers deahng with the theory of modules over valuation rings. It has been observed that a variety of concepts and basic results in abelian group theory can be extended, mutatis mutandis, to modules over valuation rings, and several other results, which require drastic modification for valuation rings, can also be dealt with by suitably generalizing ideas of abelian group theory. Our purpose is to develop techniques for modules over valuation rings; actually, these are the simplest kind of commutative non-noetherian rings. Our point of departure is abelian group theory in the local case when the groups are merely modules over Z,, the integers localized at a prime p, which is a discrete, rank one valuation domain. In the process of generalization, the most attractive and frequently used properties are sacrilied, pleasant properties we are so accustomed to in abelian groups are gone; in return, we learn new features and discover new phenomena in the behavior of modules. Several results are known on modules over valuation rings R; see the references [3-lo] or the survey article 121. This paper is devoted to the study of uniseriul R-modules, i.e., those R-modules in which the submodules form a chain. As far as the simplicity of the structure is concerned, these are second only to cyclic modules. They have been investigated by Shores and Lewis [9]; we make use of their results, especially, their description of endomorphism rings. We study various aspects of these modules with special emphasis on their quasiand pure-injectivity, as well as on the existence of pure uniserial submodules in torsion R-modules. In some cases, as expected, more explicit results can be established only under the additional hypothesis that R is almost maximal.