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摘要翻译：
给出整域$D$上一个稳定的有限型半星运算$\star$,证明了在多项式环$D[X]$上可以用规范的方法定义一个稳定的有限型半星运算$[\star]$,使得$D$是$\star$-拟PR\ufer域当且仅当$D[X]$中的每个上至零都是拟$[star]$-极大理想。这一结果完成了Houston-Malik-Mott\cite[第2节]{hmm}在star操作设置中发起的调查。此外，我们证明$D$是一个PR\ufer$\star$-乘法（例如，一个$\star$-noetherian；一个$\star$-dedekind)域当且仅当$D[X]$是一个PR\ufer$[\star]$-乘法（例如，一个$[\star]$-noetherian；一个$[\star]$-dedekind)域。作为上述方法的应用，我们利用多项式环D[X]$的乘闭集，得到了积分区域D$上有限型Gabriel-Popescu局部化系统的一个新的解释(\cg}的问题45）。
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英文标题：
《Uppers to zero and semistar operations in polynomial rings》
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作者：
Gyu Whan Chang and Marco Fontana
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Commutative Algebra 交换代数
分类描述：Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
交换环，模，理想，同调代数，计算方面，不变理论，与代数几何和组合学的联系
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.
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PDF链接：
https://arxiv.org/pdf/0706.3761
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