Let $V$ be an abelian group. Prove that if $V$ has structure of $\mathbb{Q}$-module with its given law of composition as addition, then that structure is uniquely determined.
My question is:
1) In my view, the structure is the operation on this type of algebra object, so I just have to determine $rs$ where $r \in R,s \in V$, is unique ? If my understanding is wrong, what is the meaning of STRUCTURE?
2) If (1) is correct, then my solution is to determine $\frac{p}{q}s$ is determined by the addition operation. Am I right?