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单词 Valuation ring
释义

Valuation ring

中文百科

赋值环

在抽象代数中,赋值环是一个域里的一类特别子环,可由域上的某个赋值定义。离散赋值环是其中较容易操作的一类。

赋值环是一个整环D的分式域 F的元素x,至少xxD. 一个域 F 的子环 R 被称作赋值环,若且唯若对每个  x \in F^* ,必有  x \in R x^{-1} \in R 。R被称作其分式域 F赋值环或被称作在其分式域 F的素点(位)

若 R 是主理想域,此时 R 被称为离散赋值环

英语百科

Valuation ring 赋值环

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x belongs to D.

Given a field F, if D is a subring of F such that either x or x belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring.

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更新时间:2025/6/17 23:09:50