几乎干净环与算术环
Almost clean rings and arithmetical rings
摘要：
证明了具有紧极小素谱的交换Bézout环$R$是初等除子环当且仅当对每个极小素理想$L$都是$R/L$.这个结果是通过使用环$R$的素谱的商空间$\mathrm{pSpec} R$模包含生成的等价得到的。当每一个素理想只包含一个极小素时，例如如果$R$是算术的，$\mathrm{pSpec} R$是Hausdorff的，并且在这个商空间和极小素谱$\mathrm{Min} R$之间存在一个双射，它是同胚当且仅当$\mathrm{Min} R$是紧的。如果$x$是$\mathrm{pSpec} R$的闭点，则存在纯理想$A(x)$使得$x=V(A(x))$。如果$R$几乎是干净的，即每个元素是一个正则元素与幂等元的和，它表明，$\mathrm{pSpec} R$是完全不连通的，$\forall x\in\mathrm{pSpec} R$，$R/A(x)$几乎是干净的;反之亦然，如果每个主理想都是不存在的。本文还讨论了Facchini和Faith在1995年第二届国际交换环会议上提出的一些问题。若$R$是交换环，且对于每个真理想$A$，$R/A$的子环$Q(R/A)$是IF-环，则证明了对于每个极大理想$P$，$R_P$是强离散赋值环，对于每个真理想$A$，$R/A$是半相干的.
英文摘要：
It is shown that a commutative Bézout ring $R$ with compact minimal prime spectrum is an elementary divisor ring if and only if so is $R/L$ for each minimal prime ideal $L$. This result is obtained by using the quotient space $\mathrm{pSpec} R$ of the prime spectrum of the ring $R$ modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if $R$ is arithmetical, $\mathrm{pSpec} R$ is Hausdorff and there is a bijection between this quotient space and the minimal prime spectrum $\mathrm{Min} R$, which is a homeomorphism if and only if $\mathrm{Min} R$ is compact. If $x$ is a closed point of $\mathrm{pSpec} R$, there is a pure ideal $A(x)$ such that $x=V(A(x))$. If $R$ is almost clean, i.e. each element is the sum of a regular element with an idempotent, it is shown that $\mathrm{pSpec} R$ is totally disconnected and, $\forall x\in\mathrm{pSpec} R$, $R/A(x)$ is almost clean; the converse holds if every principal ideal is finitely presented. Some questions posed by Facchini and Faith at the second International Fez Conference on Commutative Ring Theory in 1995, are also investigated. If $R$ is a commutative ring for which the ring $Q(R/A)$ of quotients of $R/A$ is an IF-ring for each proper ideal $A$, it is proved that $R_P$ is a strongly discrete valuation ring for each maximal ideal $P$ and $R/A$ is semicoherent for each proper ideal $A$.
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