Let $R$ a ring and $M$ a $R-$module. I know that $M$ is a free module if and only if $$M\cong R^{\oplus s}.$$ I know that a finitely generated module $M$ is a module s.t. there is a surjection $$R^{\oplus t}\longrightarrow M.$$
The thing, I can't describe a finitely generated module that is not a free module (I unfortunately have vector space in my mind, and I know that they are always free). Could you please give me an example of a finitely generated module that is not free ?
Take $\mathbb Z/n\mathbb Z$ as a $\mathbb Z-$module.
You will have a surjection $$\varphi:\mathbb Z\longrightarrow \mathbb Z/n\mathbb Z,$$ given by the natural homomorphism (i.e. $\varphi(n)=[n]$), but it unfortunately won't be injectif since $\ker \varphi=n\mathbb Z.$