Schwach-injektive Moduln

Summary

An <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"24"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"25"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>R$-module $M$ is called weakly injective if for every extension $M \subset X$, $M$ is coclosed in $X$. We show for every noetherian, local, one-dimensional integral domain $R$ with field of fractions $K$ and completion $\hat{R}$, that $\hat{R} \bigotimes\limits_R K$ as $\hat{R}$-module as well as $K/R$ as $R$-module are weakly injective. Moreover, we show that $K/R'$ is weakly injective iff $R$ is analytically unramified. For certain module classes over an arbitrary noetherian ring $R$ the weak injectivity can be described by means of the singular submodule $Z(M)$ as well as by the dual submodule $\overline{Z}(M)=\bigcap\,\{U \subset M\mid M/U$ is small in its injective hull$\}$ (see Talebi and Vanaja, \emph{Commun.\ Algebra} 30 (2002), 1449--1460 and Z\&quot;oschinger, \emph{Commun.\ Algebra} 33 (2005), 3389--3404). If $R$ is local and if $M$ possesses a primary decomposition we prove: every factor module of $M$ is weakly injective iff for every ${\mathfrak{p}} \in \operatorname{Coass}(M)$ the ring of fractions $R_{\mathfrak{p}}$ is a field and the fibre ring $\hat{R} \bigotimes\limits_R \kappa({\mathfrak{p}})$ is semisimple.