Einfach-teilbare und einfach-torsionsfreie R-Moduln

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 $\ne 0$, but every proper submodule $0\ne U⫋M$ is not divisible. Dually, $M$ is called simple torsion free, if $M$ ist torsion free $\ne 0$, but, for every proper submodule $0\ne U⫋M$, the factor module $M/U$ is not torsion free. Our first result is that $M\ne 0$ 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 $\mathrm{Ass}(R)$. 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 ($\mathfrak{q}\in \mathrm{Spec}(R)$), 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.

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