Modules

Let R be a ring and (M, +) an additive abelian group. Then M is called a (left) R-module if elements of M can be scalar multiplied by elements of R in a natural way. More precisely:

For each r ∈ R and m ∈ M, rm ∈ M.

Scalar multiplication by 1 ∈ R coincides with the identity map on M.

For each r ∈ R and all m, m′ ∈ M, r(m + m′) = rm + rm′.

For each m ∈ M and all r, r′ ∈ R, (r + r′)m = rm + r′m.

For each m ∈ M and all r, r′ ∈ R, (rr′)m = r(r′m).