Big indecomposable modules and direct-sum relations

Abstract

A commutative Noetherian local ring $(R,\m)$ is said to be \emph{Dedekind-like} provided $R$ has Krull-dimension one, $R$ has no non-zero nilpotent elements, the integral closure $\overline R$ of $R$ is generated by two elements as an $R$-module, and $\m$ is the Jacobson radical of $\overline R$. A classification theorem due to Klingler and Levy implies that if $M$ is a finitely generated indecomposable module over a Dedekind-like ring, then, for each minimal prime ideal $P$ of $R$, the vector space $M_P$ has dimension $0, 1$ or $2$ over the field $R_P$. The main theorem in the present paper states that if $R$ (commutative, Noetherian and local) has non-zero Krull dimension and is not a homomorphic image of a Dedekind-like ring, then there are indecomposable modules that are free of any prescribed rank at each minimal prime ideal.