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Let $n\in \mathbb Z, n\ne 0$. Prove that $ \mathbb Z/n\mathbb Z$-module $\mathbb Z/n\mathbb Z$ has a basis, but $\mathbb Z$-module $\mathbb Z/n\mathbb Z$ hasn't any basis. Hope everyone help me with that. Thanks.

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If you view $M = \mathbb Z / n \mathbb Z$ as module over $R = \mathbb Z / n \mathbb Z$ the element $1$ will form a basis: let $k \in \mathbb Z / n \mathbb Z$. Then $k = k \cdot 1$.

On the other hand, if the ring acting on $M$ is $R = \mathbb Z$ then any subset of $\mathbb Z / n \mathbb Z$ will be linearly dependent, just multiply by $n$: for example, $1 \cdot n = 0$.

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@user52523 From this answer you can see that basis elements for $\Bbb Z$ modules are in conflict with elements of finite order (torsion elements). – rschwieb Jan 14 '13 at 14:35

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