On two-generated modules over valuation domains

Abstract

Introduction. In the study of finitely generated modules over valuation domains, developed by the authors in [4], [7] and [8], a relevant share has to be assigned to indecomposable modules. Among them, the simplest examples are, apart the cyclic modules, the modules generated by two elements. Every non almost maximal valuation domain admits such indecomposable modules for which concrete constructions may be found in [1] or in [4]. In order to illustrate the contents of this paper, we need to resume here some facts on two-generated modules deriving from the above mentioned papers. Let R be a valuation domain with maximal ideal P, field of quotients Q, and let S be a maximal immediate extension of R. For general references on valuation domains and their modules we refer to [2]. Let M be a two-generated indecomposable R-module; if {xi, x2} is a system of generators, we may suppose that Ann x 1 = Ann M and that R xl is pure in M; the two ideals A = Ann x i and J = Ann (x 2 + Rxl ) are independent of the choice of the generating system, so they are determined by M. Moreover A is necessarily strictly contained in J ; A < J is called the annihilator sequence of M (see [4]). For every r ~ J, the purity of R Xl in M implies the existence of u, ~ R such that: