Review of Rings

We finish the proof that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID (we proved it’s not an euclidean ring last time). We recall the definitions and basic properties of modules, bases, generators. We start proving that if A is a PID and M is an A-module free of finite rank s, then all submodules are finitely generated and are actually free of rank smaller than s.