I was hoping someone could verify (or provide a counterexample) the following claim:
Let $R$ be a ring. Let $V$ be an abelian group, so that $V$ forms an $R$- module. If $ |R| < |V|$, then $V$ is not a $R$- module (or atmost trivial).
I know this to be true for $\mathbb{Z}^n$ -modules, but I am not sure whether this is true in general. Any help?
This is not true, even your statement about $\mathbb Z^n$ is not true. Any finite abelian group, e.g. $\mathbb Z/4\mathbb Z$ is a $\mathbb Z$-module in a natural way, but of course $|\mathbb Z|$ is much greater than $|\mathbb Z/4\mathbb Z|$.
If $R$ is a domain and you add the words "torsion-free" then it becomes true so long as $V$ is nontrivial, because then if you fix some $v\in V$, the map $r\mapsto rv$ from $R$ to $V$ is an injection.