Recently I mentioned that Noetherian rings can be characterized either by the fact that their finitely generated modules don’t have non-finitely generated submodules, or by the a priori weaker condition that their ideals are all finitely generated. In other words, if you want a counterexample to the statement that a submodule of a finitely generated module is again finitely generated, it’s sufficient to consider the case .
I also claimed that the function sending an -module to its minimum number of generators is not generally increasing: for -modules we don’t necessarily have , even when is Noetherian. For example, over , the ideal has , but .
These two facts led me to ask the following question:
If we want a commutative ring to have the property that is increasing, is it again sufficient to look for counterexamples among the ideals of ? In other words, are the rings whose ideals can all be generated by single elements (the principal ideal rings) exactly the rings with this property?