Determine all the projective ideals of the ring $\frac{\mathbb{Z}}{180\mathbb{Z}}$. Give an example of an ideal that is not projective.
We can begin by applying the CRT to find some of the projective ideals but that would not give us all of them. Then we can utilize the fact that if I is a projetive ideal there exist $\frac{\mathbb{Z}}{180\mathbb{Z}}$-module M such that
$$(\frac{\mathbb{Z}}{180\mathbb{Z}})^{(I)} \cong I \oplus M.$$
Now, if the LHS was $\frac{\mathbb{Z}}{180\mathbb{Z}}$ we could have said that I is in the form of $eR$ where $e$ is an idempotent element of the the ring $R=\frac{\mathbb{Z}}{180\mathbb{Z}}$. So, we could characterize all the projective ideals by idempotents. I don't know how to remedy the situation.