A module over a field.
| Axiom | Statement |
|---|---|
| Additive associativity | u + (v + w) = (u + v) + w |
| Additive commutativity | u + v = v + u |
| Additive identity | ∃0 ∈ V, ∀v ∈ V, v + 0 = v |
| Additive inverse | ∀v ∈ V, ∃−v ∈ V, v + (−v) = 0 |
| Multiplicative compatibility | a(bv) = (ab)v |
| Multiplicative identity | 1v = v |
| Distributivity of vector addition | a(u + v) = au + av |
| Distributivity of field addition | (a + b)v = av + bv |