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Alexandroff Topology of Algebras Over an Integral Domain
Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-02-21 , DOI: 10.1007/s00009-020-1502-z
Shai Sarussi

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:

积分域上的代数的Alexandroff拓扑

S为分数场为F的积分域,令AF代数。一个小号-subalgebra - [R称为小号-尼斯如果ř躺在在小号和本地化- [R相对于\(S {\ setminus} \ {0 \} \)。令\({\ mathbb {S}} \)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。
更新日期:2020-02-21
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