Einfach-teilbare und einfach-torsionsfreie R-Moduln

TL;DR

This paper characterizes simple torsion free and divisible modules over Noetherian local rings, linking their structure to properties of the ring's spectrum and its completions, with implications for injective hulls.

Contribution

It provides a new characterization of simple torsion free modules and analyzes conditions under which injective hulls are simple divisible, extending prior work on module structure.

Findings

Simple torsion free modules are submodules of residue fields at maximal primes.

Injective hulls are simple divisible iff the localized ring is analytically irreducible and complete.

Simple divisible modules relate to maximal ideals of the tensor product of the completion and the total quotient ring.

Abstract

Let $(R, \mathfrak{m})$ be a commutative Noetherian local ring with total quotient ring $K$. An $R$-module $M$ is called simple divisible, if $M$ is divisible $\neq 0$, but every proper submodule $0 \neq U \subsetneqq M$ is not divisible. Dually, $M$ is called simple torsion free, if $M$ ist torsion free $\neq 0$, but, for every proper submodule $0 \neq U \subsetneqq M$, the factor module $M/U$ is not torsion free. Our first result is that $M \neq 0$ is simple torsion free iff $M$ is a submodule of $\kappa(\mathfrak{p}) = R_{\mathfrak{p}}/\mathfrak{p} R_{\mathfrak{p}}$ for a maximal element $\mathfrak{p}$ in $\operatorname{Ass}(R)$. The structure of simple divisible modules is more complicated and was examined primarily by E. Matlis (1973) over 1-dimensional local $CM$-rings and by A. Facchini (1989) over any integral domain. Our main results are: If the injective hull $E(R/\mathfrak{q})$ is simple divisible ($\mathfrak{q} \in \operatorname{Spec}(R)$), then the ring $R_{\mathfrak{q}}$ is analytically irreducible and essentially complete. Especially for $\mathfrak{q} = \mathfrak{m}$, the simple divisible submodules of $E(R/\mathfrak{m})$ correspond exactly to the maximal ideals of the ring $\hat{R} \otimes_R K$, and $E(R/\mathfrak{m})$ itself is simple divisible iff $\hat{R} \otimes_R K$ is a field.