Totalseparierte Moduln

Abstract

Let $(R,\mathfrak{m})$ be a noetherian local ring, $M$ a separated $R$-module (i.e. $\bigcap _{n\ge 1}{\mathfrak{m}}^{n}M=0$) and $\hat{M}=\underleftarrow{\lim }M/{\mathfrak{m}}^{n}M$ its completion. Generally, $M$ is not pure in $\hat{M}$ and $\hat{M}$ 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 $\hat{M}$ and, under additional conditions, $\hat{M}$ 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 $\hat{M}$ are totally separated and establish a close connection to the class of strictly pure-essential extensions. In section 3, we replace the completion $\hat{M}$ 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 $A={\bigcap }_{n\ge 1}(M+{\mathfrak{m}}^{n}P)$. 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 $A/M$.

12 pages, in German