Mariano's answer is perfect, of course, but let me add a few more details and generalities.
You can view a ring with unit as a (universal) algebra $\langle R, +, \cdot, -, 0, 1\rangle$, and you can view a module over this ring as the algebra $\langle M, +, -, 0, \{f_r : r\in R\}\rangle$, where the reduct $\langle M, +, -, 0\rangle$ is an abelian group, each of the unary operations $f_r$ (scalar multiplication by $r$) is an endomorphism of this group, and the map $r \mapsto f_r$ is a ring homomorphism. If the ring $R$ happens to be a field, we call $M$ a vector space.
The best reference (imho) for this view of the world is "Algebras, Lattices, Varieties, Vol. 1" by McKenzie, McNulty, Taylor.