论文标题
分配FCP扩展
Distributive FCP extensions
论文作者
论文摘要
我们正在处理$ r \ subseteq s $的扩展，其poset $ [r，s]的链条是有限的（{\ em i.e.} $ r \ subseteq s $具有fcp属性），并且$ [r，s] $是分配latte latte latte latte latte latte lattens expressions extens fcp fcp fcp fcp。请注意，分配FCP扩展的晶格$ [R，S] $是有限的。本文是我们较早的论文的延续，我们研究了Catenarian和Boolean扩展。实际上，对于FCP扩展而言，以下含义存在：布尔$ \ rightarrow $分配$ \ rightarrow $ catenarian。实际上，分布FCP扩展的全面表征实际上仍然是一个挑战，本质上是因为未完全解决了场扩展的相同问题。然而，我们能够为某些类别的扩展表现出很多积极的结果。一个主要结果是，当且仅当$ r \ r \ subseteq \ overline r $是分布式时，FCP扩展名$ r \ subseteq s $是分布式的，其中$ \ overline r $是$ r $ in $ s $的整体关闭。特别注意分布场扩展。
We are dealing with extensions of commutative rings $R\subseteq S$ whose chains of the poset $[R,S]$ of their subextensions are finite ({\em i.e.} $R\subseteq S$ has the FCP property) and such that $[R,S]$ is a distributive lattice, that we call distributive FCP extensions. Note that the lattice $[R,S]$ of a distributive FCP extension is finite. This paper is the continuation of our earlier papers where we studied catenarian and Boolean extensions. Actually, for an FCP extension, the following implications hold: Boolean $\Rightarrow$ distributive $\Rightarrow$ catenarian. A comprehensive characterization of distributive FCP extensions actually remains a challenge, essentially because the same problem for field extensions is not completely solved. Nevertheless, we are able to exhibit a lot of positive results for some classes of extensions. A main result is that an FCP extension $R\subseteq S$ is distributive if and only if $R\subseteq\overline R$ is distributive, where $\overline R$ is the integral closure of $R$ in $S$. A special attention is paid to distributive field extensions.
