Review of Rings

We finish 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. Then we prove that (still over a PID) all finitely generated modules are direct sum of cyclic modules. The primary decomposition theorem. Vector spaces over k as k[x]-modules. Application to rational normal form and Jordan form of a matrix.

References: Notes by Prof. Giudetti, notes by Prof. De Stefano, notes by Prof. Valla