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分式环

抽象代数中,分式环分式域是包含一个整环的最小,典型的例子是有理数域之于整数环。此外分式环也可以推广到一般的交换环,此时通常称作全分式环

分式环有时也被称为商域,但此用语易与商环混淆。

构造

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分式环是局部化的一个简单特例。以下设 为一个整环,而

在集合 上定义下述等价关系

等价类 可以想成“分式” ,上述等价关系无非是推广有理数的通分;借此类比,在商集 上定义加法与乘法为:

可验证上述运算是明确定义的。此外还有环同态 ,定义为 ;这是一个单射。于是可定义分式环 ,再配上上述的加法与乘法运算。在实践上,我们常迳将 里的元素写作分式

泛性质

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整环 的分式环 及其自然环同态 满足以下的泛性质

对任何环 及环同态 ,若 中的元素在 下的像皆可逆,则存在唯一的环同态 ,使得 的合成。

此性质不外是形式地表达了“K(R) 是包含 R 的最小的域”这个陈述。据此泛性质可形式地证明:任何一组资料 若使得 中的元素在 下的像皆可逆,且满足上述泛性质,则 必与 同构。

例子

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推广

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对于一般的交换环 (容许有零因子),分式环是一种退而求其次的建构:我们想找使 为单射的“最大”局部化,详述如下:

中的非零因子所成子集,它是个积性子集,因此可对之作局部化。令 ,此时 常被称作 全分式环

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分式环
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