As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem (i.e., ACC iff every submodule is finitely generated iff maximal submodule condition) and the exact sequences theorem which is demonstrated in Atiyah and MacDonald's book, from which all basic results follow as corollaries.
After showing all this, every source I've consulted either stops talking about them or immediately jumps to noetherian and artinian rings. In both cases, applications aren't discussed. This led me to the natural question,
Why are noetherian and artinian modules important?
I'd love to show the students some applications or examples. Someone even mentioned there is a relation between some remainder theorem (chinese?) and these concepts, is this true?
There's an almost identical question which only deals with the rings case. The most voted answer to that question basically says that noetherian and artinian rings aren't that important and that usually the hypothesis of a ring being either noetherian or artinian can be weakened. I'm not looking for this kind of answers and I would even appreciate non-trivial examples of stronger theorems which also hold for this modules.