About the basis of a module

Question

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.

Answer 1

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, $1 \cdot n = 0$.