A module over a field.
| Axiom | Statement |
|---|---|
| Additive associativity | \(\mathbf u+(\mathbf v+\mathbf w)=(\mathbf u+\mathbf v)+\mathbf w\) |
| Additive commutativity | \(\mathbf u+\mathbf v=\mathbf v+\mathbf u\) |
| Additive identity | \(\exists\mathbf 0\in V,\forall\mathbf v\in V,\mathbf v+\mathbf 0=\mathbf v\) |
| Additive inverse | \(\forall\mathbf v\in V,\exists-\mathbf v\in V,\mathbf v+(-\mathbf v)=\mathbf 0\) |
| Multiplicative compatibility | \(a(b\mathbf v)=(ab)\mathbf v\) |
| Multiplicative identity | \(1\mathbf v=\mathbf v\) |
| Distributivity of vector addition | \(a(\mathbf u+\mathbf v)=a\mathbf u+a\mathbf v\) |
| Distributivity of field addition | \((a+b)\mathbf v=a\mathbf v+b\mathbf v\) |