module (original) (raw)

Let R be a ring with identity. A left module M over R is a set with two binary operations, +:M×M⟶M and ⋅:R×M⟶M, such that

1. 1. (𝐮+𝐯)+𝐰=𝐮+(𝐯+𝐰) for all 𝐮,𝐯,𝐰∈M
2. 1. 𝐮+𝐯=𝐯+𝐮 for all 𝐮,𝐯∈M
3. 1. There exists an element 𝟎∈M such that 𝐮+𝟎=𝐮 for all 𝐮∈M
4. 1. For any 𝐮∈M, there exists an element 𝐯∈M such that 𝐮+𝐯=𝟎
5. 1. a⋅(b⋅𝐮)=(a⋅b)⋅𝐮 for all a,b∈R and 𝐮∈M
6. 1. a⋅(𝐮+𝐯)=(a⋅𝐮)+(a⋅𝐯) for all a∈R and 𝐮,𝐯∈M
7. 1. (a+b)⋅𝐮=(a⋅𝐮)+(b⋅𝐮) for all a,b∈R and 𝐮∈M

A left module M over R is called unitary or unital if 1R⋅𝐮=𝐮 for all 𝐮∈M.