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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)?

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up vote 2 down vote accepted

For (4): Consider $\Bbb{Z} \subseteq \Bbb{Q}$ and consider $\Bbb{Z}$ as a module over itself (i.e. $\Bbb{Z}= S$ and $\Bbb{Q} = R$). Can you make $\Bbb{Z}$ into a $\Bbb{Q}$ - module?

Can you use this to give a counterexample to (3)?

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@aliakbar Confused about what? Please tell me as I can't read your mind. –  fpqc Feb 3 '13 at 11:30
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