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Let $R$ be a commutative Noetherian local ring, $\mathfrak{G}$ a Gabriel topology on $R$, and $\mathfrak{G}^\ast$ the set of all maximal elements of Spec($R)\backslash \mathfrak{G}$. We determine all simple $\mathfrak{G}$-torsion free $R$-modules $M$, as well as all simple $\mathfrak{G}$-divisible Artinian $R$-modules $N$. A central role is played by the set $\mathfrak{G}^{\ast}$: Its elements correspond exactly to the maximal ideals of the quotient ring $R_{\mathfrak{G}}$, if $\mathfrak{G}$ is perfect., Comment: in German

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. Let $\mathcal{P}$ be the class of all $I$-generated $R$-modules $M$ (i.e. there is an epimorphism $I^{(\Lambda)} \twoheadrightarrow M$) and let $\mathcal{S}$ be the class of all $I^{\circ}$-cogenerated $R$-modules $N$ (i.e. there is a monomorphism $N \hookrightarrow (I^{\circ})^{\Lambda}$ with $I^{\circ} = \operatorname{Hom}_R(I,E)$). We give a complete description of all injective and flat modules in $\mathcal{P}$ and $\mathcal{S}$. We show that $(\mathcal{S},\mathcal{P})$ forms a dual pair in the sense of Mehdi--Prest(2015) and that $\mathcal{P}$ is always closed under pure submodules. We determine all ideals $I$ for which $\mathcal{P}$ is closed under submodules, $\mathcal{S}$ is closed under factor modules and $\mathcal{P}$ (resp. $\mathcal{S}$) is closed under group extensions. In the last section, we examine the submodules $\gamma(M) = \sum\{U \subset M \,|\, U \in \mathcal{P}\}$ and $\kappa(M) = \bigcap \{V \subset M \,|\, M/V \in \mathcal{S}\}$ for all $R$-modules $M$, and we specify their explicit structure in special cases., Comment: in German

Let $(R, \mathfrak m)$ be a commutative noetherian local ring. We investigate under which conditions an $R$-module $M$ is generated by an ideal $I$, i.e. there exists an epimorphism $I^{(\Lambda)} \twoheadrightarrow M$. If $M$ is uniserial, i.e. $\mathcal{L}(M)$ is totally ordered and finite, this is equivalent to $\mathfrak{m}^{n-1} \cdot I \not\subset \operatorname{Ann}_R(M) \cdot I$ ($\operatorname{length}(M) = n \geq 1$). If $M$ is cyclic and $I = \mathfrak{m}$, this is equivalent to: Either it is $M \cong R/\mathfrak{p}$ ($R/\mathfrak{p}$ a discrete valuation ring) or $M \cong C/\operatorname{So}(C)$ ($C$ a uniserial $R$-module). If $A$ is free and $B$ is a submodule of $A$, then the Matlis dual $(A/B)^{\circ} = operatorname{Hom}_R(A/B, E)$ is $I$-generated if and only if $B = (IB) :_A I$. In the case $I = \mathfrak{m}$, this condition leads to the "basically full ideals" considered by Heinzer, Ratliff~Jr. and Rush. By studying the dual condition $M = I(M :_X I)$ in the last section, we can generalize some results of that work., Comment: 9 pages, in German

Let (R,m) be a noetherian local ring and let $\mathcal{C}$ be the class of all R-modules M which possess a reflexive submodule U such that M/U is finitely generated. For every R-module $M\in \mathcal{C}$ the canonical embedding $\varphi: M\to M^{oo}$ is pure-essential. We investigate in the first section under which conditions the reverse is true, for example if R is a discrete valuation ring or if R does not have nilpotent elements and M is flat. In section 2 we determine all reflexive and flat R-modules with the help of a certain analogy between the localization $R_q$ and the injective hull of R/q. In section 3 we show: If the property 'pure-essential' is transitive for a domain R, then it follows that $dim(R)\leq 1$., Comment: 11 pages, in German