criterion for a module to be noetherian
Suppose is Noetherian (over a ring ), and a submodule. Since any submodule of is finitely generated, any submodule of , being a submodule of , is finitely generated as well. Next, if is a submodule of , and if is a generating set for , then is a generating set for . Conversely, if every submodule of is Noetherian, then , being a submodule itself, must be Noetherian. ∎
A weaker form of the converse is the following:
If is a submodule of such that and are Noetherian, then is Noetherian.
Suppose is an ascending chain of submodules of . Let , then is an ascending chain of submodules of . Since is Noetherian, the chain terminates at, say . Let , then is an ascending chain of submodules of . Since is Noetherian, the chain stops at, say . Let . Then we have and . We want to show that . Since , we need the other inclusion. Pick . Then , where . This means that . But as well, so . Since , this means that or . ∎