域上的代数

定义

例1：复数

• ${\displaystyle (\mathbf {x} +\mathbf {y} )\mathbf {z} =\mathbf {x} \mathbf {z} +\mathbf {y} \mathbf {z} }$
• ${\displaystyle (a\mathbf {x} )(b\mathbf {y} )=(ab)(\mathbf {x} \mathbf {y} )}$

例2：四元数

{\displaystyle {\begin{aligned}&(a+b\mathbf {i} +c\mathbf {j} +d\mathbf {k} )(w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} )\\=&(aw-bx-cy-dz)+(ax+bw+cz-dy)\mathbf {i} +(ay-bz+cw+dx)\mathbf {j} +(az+by-cx+dw)\mathbf {k} \end{aligned}}}

${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {q} }$ ${\displaystyle \mathbf {r} }$  为任意四元数而 ${\displaystyle a}$ ${\displaystyle b}$  为任意实数，可验证：

• ${\displaystyle \mathbf {p} (\mathbf {q} +\mathbf {r} )=\mathbf {p} \mathbf {q} +\mathbf {p} \mathbf {r} }$
• ${\displaystyle (\mathbf {p} +\mathbf {q} )\mathbf {r} =\mathbf {p} \mathbf {r} +\mathbf {q} \mathbf {r} }$
• ${\displaystyle (a\mathbf {p} )(b\mathbf {q} )=(ab)(\mathbf {p} \mathbf {q} )}$

例3：三维向量的叉积

• ${\displaystyle \mathbf {u} \times (\mathbf {v} +\mathbf {w} )=\mathbf {u} \times \mathbf {v} +\mathbf {u} \times \mathbf {w} }$
• ${\displaystyle (\mathbf {u} +\mathbf {v} )\times \mathbf {w} =\mathbf {u} \times \mathbf {w} +\mathbf {v} \times \mathbf {w} }$
• ${\displaystyle (a\mathbf {p} )\times (b\mathbf {q} )=(ab)(\mathbf {p} \times \mathbf {q} )}$

定义

• 对加法的左分配律：${\displaystyle \mathbf {x} \times (\mathbf {y} +\mathbf {z} )=\mathbf {x} \times \mathbf {y} +\mathbf {x} \times \mathbf {z} }$
• 对加法的右分配律：${\displaystyle (\mathbf {x} +\mathbf {y} )\times \mathbf {z} =\mathbf {x} \times \mathbf {z} +\mathbf {y} \times \mathbf {z} }$
• 与标量乘法相容：${\displaystyle (a\cdot \mathbf {x} )\times (b\cdot \mathbf {y} )=(ab)\cdot (\mathbf {x} \times \mathbf {y} )}$