Showing that $\mathbb{Z}/3\mathbb{Z}$ is a projective $\mathbb{Z}/12\mathbb{Z}$-module but not a free $\mathbb{Z}/12\mathbb{Z}$-module

Question

I am trying to show that $\mathbb{Z}/3\mathbb{Z}$ is a projective $\mathbb{Z}/12\mathbb{Z}$-module but not a free $\mathbb{Z}/12\mathbb{Z}$-module.

So, far I have been able to show that $4(\mathbb{Z}/12\mathbb{Z})\cong \mathbb{Z}/3\mathbb{Z}$ and $3(\mathbb{Z}/12\mathbb{Z})\cong \mathbb{Z}/4\mathbb{Z}$. Moreover, $\mathbb{Z}/3\mathbb{Z}\bigcap \mathbb{Z}/4\mathbb{Z}=\{0\}$. So that $\mathbb{Z}/12\mathbb{Z}\cong \mathbb{Z}/3\mathbb{Z}\bigoplus \mathbb{Z}/4\mathbb{Z}$.

Now, I want an easy way of seeing that $\mathbb{Z}/3\mathbb{Z}$ is not a free on $\mathbb{Z}/12\mathbb{Z}$.

Also, that $\mathbb{Z}/3\mathbb{Z}$ is a projective $\mathbb{Z}/12\mathbb{Z}$-module.

Answer 1

Hint:

Free modules over $R$ look like $\oplus_{i\in I}R$.

Can't you just count the elements of such a module when $I$ is a finite set?