Totalseparierte Moduln

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 $\widehat{M}=\underleftarrow{\lim }M\mathrm{/}{\mathfrak{m}}^{n}M$ 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. $X\underset{R}{\otimes} M$isseparatedforallfinitelygenerated$R$-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 $M = \coprod_{i\in I}M_i$withthe$\mathfrak{m}$-adicclosure$A$of$M$in$P = \prod_{i\in I} M_i$,i\mathrm{.}e\mathrm{.}with$A = \bigcap_{n \geq 1}(M + \mathfrak{m}^n P)$. We give criteria so that $A\mathrm{/}M$ is radical and show that this always holds in the countable case $M={\coprod }_{i=1}^{\mathrm{\infty }}{M}_i$. Finally, we deal with the case that $A$ is even totally separated and additionally determine the coassociated prime ideals of $A\mathrm{/}M$.