arXiv math/06115562006-11math.AC

关于整环的$t$ -连接覆盖环集

On the Set of $t$-Linked Overrings of an integral domain

Abdeslam Mimouni

备注：: 14 pages

摘要：

设$R$是具有商域$L$整环.如果$I^{-1}=R$意味着$(T:IT)=T$对于每个环生成$R$的理想$I$，则$R$的上环$T$是$t$在$R$上连接的。令$O_{t}(R)$表示$R$的所有$t$连接的覆盖环的集合，并且$O(R)$表示$R$的所有覆盖环的集合。本文研究了集合$O_{t}(R)$上的一些有限性条件。特别地，我们证明了：如果$O_{t}(R)$是有限的，那么$O(R)$和$O_{t}(R)=O(R)$也是有限的;如果$R$的每个$t$ -连接的覆盖环链是有限的，那么$R$的每个覆盖环链也是有限的.这产生了$t$连接的方法比Gilmer在\cite{G1}中的处理更有效。我们还研究了有限性条件在一些Noether的设置，如Mori域，拟同调Mori域，Krull域等，我们建立了一个连接之间的$O_{t}(R)$和所有强除理想的集合$R$，我们得出的结论是由域$R$是$t$ -连接下的所有覆盖环的特征。

英文摘要：

et $R$ be an integral domain with quotient field $L$. An overring $T$ of $R$ is $t$-linked over $R$ if $I^{-1}=R$ implies that $(T:IT)=T$ for each finitely generated ideal $I$ of $R$. Let $O_{t}(R)$ denotes the set of all $t$-linked overrings of $R$ and $O(R)$ the set of all overrings of $R$. The purpose of this paper is to study some finiteness conditions on the set $O_{t}(R)$. Particularly, we prove that if $O_{t}(R)$ is finite, then so is $O(R)$ and $O_{t}(R)=O(R)$, and if each chain of $t$-linked overrings of $R$ is finite, then each chain of overrings of $R$ is finite. This yields that the $t$-linked approach is more efficient than the Gilmer's treatment in \cite{G1}. We also examine the finiteness conditions in some Noetherian-like settings such as Mori domain, quasicoherent Mori domain, Krull domain etc. We establish a connection between $O_{t}(R)$ and the set of all strongly divisorial ideals of $R$ and we conclude by a characterization of domains $R$ that are $t$-linked under all their overrings.

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