# 相反環

## 例子

### 兩個生成元嘅自由代數

${\displaystyle k}$ 上面嘅雙生成元自由代數${\displaystyle k\left\langle x,y\right\rangle }$ 上面嘅乘法係用字串連接嚟定義嘅，例如：

${\displaystyle (x+y)(2x+1)=2x^{2}+x+2yx+y}$

### 四元數代數

${\displaystyle k}$ 上面嘅四元數代數（quaternion algebra）[3]${\displaystyle H(a,b)}$ 係一個除代數，有三個生成元${\displaystyle i,j,k}$ 同埋三個關係

${\displaystyle i^{2}=a}$ , ${\displaystyle j^{2}=b}$ , 同 ${\displaystyle k=ij=-ji}$

${\displaystyle x=x_{0}+x_{i}i+x_{j}j+x_{k}k}$

${\displaystyle \cdot }$  ${\displaystyle i}$  ${\displaystyle j}$  ${\displaystyle k}$
${\displaystyle i}$  ${\displaystyle a}$  ${\displaystyle k}$  ${\displaystyle aj}$
${\displaystyle j}$  ${\displaystyle -k}$  ${\displaystyle b}$  ${\displaystyle -bi}$
${\displaystyle k}$  ${\displaystyle -aj}$  ${\displaystyle bi}$  ${\displaystyle -ab}$

${\displaystyle *}$  ${\displaystyle i}$  ${\displaystyle j}$  ${\displaystyle k}$
${\displaystyle i}$  ${\displaystyle a}$  ${\displaystyle -k}$  ${\displaystyle -aj}$
${\displaystyle j}$  ${\displaystyle k}$  ${\displaystyle b}$  ${\displaystyle bi}$
${\displaystyle k}$  ${\displaystyle aj}$  ${\displaystyle -bi}$  ${\displaystyle -ab}$

## 性質

• 兩個環${\displaystyle R_{1},R_{2}}$ 同構若且唯若佢哋分別嘅相反環同構。
• 一個環嘅相反環嘅相反環係返佢自己。
• 一個環同佢嘅相反環係反同構嘅。
• 一個環係交換環若且唯若佢嘅乘法同反乘法係一樣嘅[2]
• 一個環嘅左理想正正就係相反環嘅右理想[4]
• 除環嘅相反環都係除環[5]
• 一個環上面嘅左模就係相反環上面嘅右模，反之亦然[6]

## 參考

1. Berrick & Keating (2000), p. 19
2. Bourbaki 1989, p. 101.
3. Milne. Class Field Theory. p. 120.
4. Bourbaki 1989, p. 103.
5. Bourbaki 1989, p. 114.
6. Bourbaki 1989, p. 192.