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可逆元

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单位又被称为可逆元。在数学里,于一(有单位的) 内的可逆元是指一 的可逆元,即一元素 使得存在一于 内的 有下列性质: ,其中 是乘法单位元

亦即, 内乘法幺半群的一可逆元素。

可逆元群

的可逆元组成了一于乘法下的 ,称做 可逆元群。可逆元群U(R)有时亦被标记成R*R×

在一可交换单作环R内,可逆元群U(R)以乘法作用R上头。此一作用的轨道(orbit)被称为结合集合;换句话说,存在一于R上的等价关系 ~ ,且当r~s时,表示存在一可逆元u使得r=us

U是一由环范畴至群范畴的函子:每一个环同态 f : RS 都可导出一群同态U(f) : U(R) → U(S),当f会将可逆元映射至可逆元时。此一函数子有为整数群环结构的左伴随。

一个环R是一个除环当且仅当R* = R \ {0}。

例子

  • 整数环里,可逆元为±1。其每一轨道内都有两个元素n和−n
  • 任一单位根均是某一单作环内的可逆元。(若是一单位根,且,则亦为的元素)。
  • 代数数论里,狄利克雷单位定理证明了许多代数整数环内可逆元的存在域。例如,在环,因此都是可逆元。
  • 在环,于一上的矩阵内,其可逆元恰好就是可逆矩阵
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可逆元
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