We say that an $A$-module $M$ is Noetherian if all of its submodules are finitely generated. Having that definition in mind can anyone give me some hints to prove that if $M$ and $N$ are Noetherian modules then $M \oplus_A N$ must be Noetherian?
The problem I'm having is that an arbitrary submodule $Q$ of $M \oplus_A N$ is not necessarly of the form $Q = \bar{M} \oplus \bar{N}$ where $\bar{M}$ and $\bar{N}$ are submodules of $M$ and $N$ respectively. So I am out of ideas, and will be very happy to recieve some help.
Best Answer
Hint: first show that if $P \subseteq Q$ are $A$-modules, then $P, Q/P$ Noetherian implies $Q$ Noetherian.