Why do we define modules only on additive groups?

Question

By definition, modules are additive abelian groups that satisfy a few properties. I know why abelian property is necessary as it is forced by its axioms. But why can we do the same with multiplicative groups, like $R^*$ or $S^1$ (mult group of roots of unity) or circle group etc.

Answer 1

The notational reason is that we introduce a new operation for modules: multiplication by integers. In order to avoid confusion with the old operation, belonging to the abelian group, we denote the old operation by addition.

These two operations do come with a distributive property, so they behave like addition and multiplication "should" together.

And - well, in the big picture, if we have an abelian group, it doesn't matter what we call that group's operation. We could call the circle group's operation addition if we felt like it - it is addition for angles, after all. What we call the operation only matters if we're embedding the group in some structure with more operations, like putting the circle group into the field $\mathbb{C}$ as the elements with absolute value $1$.

Answer 2

Well, you could take the set of positive real numbers and define $x+y = x*y$ (addition) and $cx=x^c$ (scalar multiplication). This gives you a vector space. (You basically consider the logarithm).