arXiv subjects

Helmut Zöschinger

Publications and source records attributed to Helmut Zöschinger.

8 recordsLinked to original sources

arXiv2019

Einfach-teilbare und einfach-torsionsfreie R-Moduln

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 $κ(\mathfrak{p}) = R_{\mathfrak{p}}/\mathfrak{p} R_{\mathfrak{p}}$ 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.

math.AC↗

arXiv2018

Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln III

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. For every $R$-module $M$, $γ_I(M) = \sum\{ \operatorname{Bi} f \,|\, f \in \operatorname{Hom}_R(I,M)\}$ is called the trace of $I$ in $M$. It is easy to see that $\operatorname{Ext}_R^1(R/I,M) = 0$ always implies $IM = γ_I(M)$. If the second condition holds for all ideals $I$ of $R$, we say that $M$ is excellent. In part 1, we show a number of conditions for these modules, which are well-known for injective modules. In the second part, we examine the special case $M = R$. In particular, we show that for every prime ideal $\mathfrak{p}$ the equality $\mathfrak{p} = γ_{\mathfrak{p}}(R)$ holds iff $R_{\mathfrak{p}}$ is not a discrete valuation ring. From the results by Matlis (1973) about 1-dimensional local CM-rings and with the help of the first neighborhood ring $Λ$, it follows immediately that $γ_{\mathfrak{m}^n} (R) = Λ^{-1}$ for almost all $n \geq 1$. In the third part, we examine the dual construction $κ_I(M) = \bigcap \{ \operatorname{Ke} f \,|\, f\in \operatorname{Hom}_R(M,I^\circ) \}$ and reduce the main results about $\operatorname{Tor}_1^R(M, R/I) = 0$ and $κ_I(M) = M[I]$ to part 1 by considering the Matlis dual $M^\circ = \operatorname{Hom}_R(M, E)$ and the equalities $γ_I(M^\circ) = \operatorname{Ann}_{M^\circ}(κ_I(M))$, $κ_I(M^\circ) = \operatorname{Ann}_{M^\circ}(γ_I(M))$.

math.AC↗

arXiv2017

Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln II

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 $I^{(Λ)} \twoheadrightarrow M$) and let $\mathcal{S}$ be the class of all $I^{\circ}$-cogenerated $R$-modules $N$ (i.e. there is a monomorphism $N \hookrightarrow (I^{\circ})^Λ$ 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 $γ(M) = \sum\{U \subset M \,|\, U \in \mathcal{P}\}$ and $κ(M) = \bigcap \{V \subset M \,|\, M/V \in \mathcal{S}\}$ for all $R$-modules $M$, and we specify their explicit structure in special cases.

math.AC↗

arXiv2016

Über die von einem Ideal $I \subset R$ erzeugten $R$-Moduln

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 $I^{(Λ)} \twoheadrightarrow M$. 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.

math.AC↗

arXiv2015

Totalseparierte Moduln

Let $(R, \mathfrak{m})$ be a noetherian local ring, $M$ a separated $R$-module (i.e. $\bigcap\limits_{n\geq 1}\mathfrak{m}^n M = 0$) and $\widehat{M} = \lim\limits_{\leftarrow} M/\mathfrak{m}^n M$ its completion. Generally, $M$ is not pure in $\widehat{M}$ and $\widehat{M}$ is not pure-injective. But if $M$ is totally separated, i.e. $X\underset{R}{\otimes} M$ is separated for all finitely generated $R$-modules $X$, the situation improves: In this case, $M$ is pure in $\widehat{M}$ and, under additional conditions, $\widehat{M}$ is even pure-injective, e.g. if $M\cong X^{(I)}$ holds with $X$ finitely generated or $M \cong\coprod_{i=1}^{\infty} R/\mathfrak{m}^i$. In section 2, we investigate the question under which conditions both $M$ and $\widehat{M}$ are totally separated and establish a close connection to the class of strictly pure-essential extensions. In section 3, we replace the completion $\widehat{M}$ in the case $M = \coprod_{i\in I}M_i$ with the $\mathfrak{m}$-adic closure $A$ of $M$ in $P = \prod_{i\in I} M_i$, i.e. with $A = \bigcap_{n \geq 1}(M + \mathfrak{m}^n P)$. We give criteria so that $A/M$ is radical and show that this always holds in the countable case $M = \coprod_{i=1}^{\infty} M_i$. Finally, we deal with the case that $A$ is even totally separated and additionally determine the coassociated prime ideals of $A/M$.

math.AC↗

arXiv2014

Über rein-wesentliche Erweiterungen

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 $φ: M\to M^{oo}$ 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 $dim(R)\leq 1$.

math.AC↗

arXiv2013

Eine Charakterisierung der Matlis-reflexiven Moduln

Let $(R,\my)$ be a noetherian local ring, $E$ the injective hull of $k=R/\my$ and $M^\circ=$ Hom$_R(M,E)$ the Matlis dual of the $R$-module $M$. If the canonical monomorphism $φ: M \to \moo$ is surjective, $M$ is known to be called (Matlis-)reflexive. With the help of the Bass numbers $μ(\py,M)=\dim_{κ(\py)}($Hom$_R(R/\py,M)_\py)$ of $M$ with respect to $\py$ we show: $M$ is reflexive if and only if $μ(\py,M)=μ(\py,\moo)$ for all $\py \in $ Spec$(R)$. From this it follows for every $R$-module $M$: If there exists a monomorphism $\moo \hookrightarrow M$ or an epimorphism $M \twoheadrightarrow \moo$, then $M$ is already reflexive.

math.AC↗

arXiv2012

Über die assoziierten Primideale der Vervollständigung

Let $(R,\my)$ be a noetherian local ring and let $M$ be an $R$-module such that $\bigcap\limits_{n\geq 1} \my^n M=0.$ Let $\hat{M}$ be the completion of $M$. We show that Ass$(\hat{M})=$ Koatt$(M)$ holds in the following three cases: if $\dim(R)\leq 1,$ if $\hat{M}$ as $R$-module is flat, or if $M$ is the direct sum of $R$-modules which are finitely generated. If $M$ is pure in $\hat{M}$ then at least Ass$(\hat{M}) \subset $ Koatt$(M)$ holds. If the conjecture by A.-M.Simon on complete $R$-modules is valid then one has Koatt$(M)\subset $ Ass$(\hat{M}).$

math.AC↗