I need to prove the following fact: Let R be a unital ring, M is a finitely generated R-module. Submodules of a finitely generated $R$-module is contained in a maximal ideal. I found the following post that deals with the same problem.
However, I don't quite understand the proof given:
Why we can be sure that the submodules of the the $R$-module forms a chain? My interpretation of chain is that I have $N_1 \subseteq N_2 \subseteq .....$. What happens if some of the submodules is not finitely generated? Am I misunderstanding the definition of chain?
Am I correct to say that the maximal submodule is just the union of all the submodules of the $R$-module?