Finitely generated mixed modules of Warfield type

Abstract

Let $R$ be a local one-dimensional domain, with maximal ideal $\mathfrak{M}$, which is not a valuation domain. We investigate the class of the finitely generated mixed $R$-modules of Warfield type, so called since their construction goes back to R.B. Warfield. We prove that these $R$-modules have local endomorphism rings, hence they are indecomposable. We examine the torsion part $t(M)$ of a Warfield type module $M$, investigating the natural property $t(M)\subset \mathfrak{M}M$. This property is related to $b/a$ being integral over $R$, where $a$ and $b$ are elements of $R$ that define $M$. We also investigate $M/t(M)$ and determine its minimum number of generators.