Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R is lying over S and the localization of R with respect to \(S {\setminus } \{ 0 \}\) is A. Let \({\mathbb {S}}\) be the set of all S-nice subalgebras of A. We define a notion of open sets on \({\mathbb {S}}\) which makes this set a \(T_0\)-Alexandroff space. This enables us to study the algebraic structure of \({\mathbb {S}}\) from the point of view of topology. We prove that an irreducible subset of \({\mathbb {S}}\) has a supremum with respect to the specialization order. We present equivalent conditions for an open set of \(\mathbb S\) to be irreducible, and characterize the irreducible components of \({\mathbb {S}}\). We also characterize quasi-compactness of subsets of a \(T_0\)-Alexandroff topological space.

中文翻译：

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积分域上的代数的Alexandroff拓扑

令S为分数场为F的积分域，令A为F代数。一个小号-subalgebra - [R的甲称为小号-尼斯如果ř躺在在小号和本地化- [R相对于\（S {\ setminus} \ {0 \} \）是甲。令\（{\ mathbb {S}} \）为A的所有S -nice子代数的集合。我们在\（{\ mathbb {S}} \）上定义一个开放集的概念，这使该集合成为\（T_0 \）-亚历山大空间。这使我们能够从拓扑的角度研究\（{\ mathbb {S}} \\）的代数结构。我们证明\（{\ mathbb {S}} \）的不可约子集在专业化顺序方面具有至高无上的地位。我们给出一个开放集合\（\ mathbb S \）不可约的等价条件，并刻画\（{\ mathbb {S}} \）的不可约成分。我们还表征了\（T_0 \）- Alexandroff拓扑空间的子集的拟紧性。