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赋值环.
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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。
定义
赋值环是一个整环D,满足其分式域 F的任一非零元素x,至少有x 或 x −1 ∈ D.
一个域 F 的子环 R 被称作赋值环,当且仅当对每个
,必有
或
。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。
性质
- 令
,则
是 F 中唯一的极大理想。
- 承上,
被称作 R 的剩余域。
文献
- Nicolas Bourbaki, Algèbre commutative, Chapitre 5, 6: entiers ; valuations (1964), Eléments de mathématique, P. A. Hermann.
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