I am trying to understand the module axioms.
Below should be the standard module axioms taken from Artin's 'Algebra':
Let R be a ring. An R-module V is an abelian group with law of composition written +, together with a scalar multiplication $R \times V \longrightarrow V$, written $r, v \mapsto rv$, which satisfies
- $1v = v$,
- $(rs)v = r(sv)$,
- $(r+s)v = rv + sv$,
- $r(v+v') = rv + rv'$.
According to the axioms, though it sounds wrong, this construction is legal, right?: Let $V := \mathbb{R}$. Then V is a $\mathbb{Q}$-module because it satisfies all above axioms with the scalar multiplication by $q \in \mathbb{Q}$. If this is wrong, what is wrong?