Let $R$ be a nonzero commutative ring with identity. Which of the following statements is true?
1) Every $R$-module is a vector space over $R$.
2) Every $R$-module has a $\mathbb Z$-module structure.
3) Every $\mathbb Z$-module has a nontrivial $R$-module structure.
4) If $S$ is a subring of $R$, then every $S$-module has a $R$-module structure.
It is clear that (1) is true if and only if $R$ is a field. (2) is true. But (3) and (4)?