Interlude: principal ideal rings

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 M is again finitely generated, it’s sufficient to consider the case M = R.

I also claimed that the function \mu_R sending an R-module to its minimum number of generators is not generally increasing: for R-modules N\subseteq M we don’t necessarily have \mu_R(N)\leq \mu_R(M), even when R is Noetherian. For example, over R = \mathbb{C}[x,y], the ideal I = (x,y)\subseteq R has \mu_R(I)=2, but \mu_R(R) = 1.

These two facts led me to ask the following question:

If we want a commutative ring R to have the property that \mu_R is increasing, is it again sufficient to look for counterexamples among the ideals of R? 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?

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What is a Noetherian ring?

Noetherianness is about making sure small modules don’t have big parts. But how do you measure how big a module is? Remember that while every vector space has a dimension—the size of a basis—most modules don’t have a basis. We can still talk about a generating set for an R-module M, though, like \{1, 1/2, 1/3, \dots\} for the \mathbb{Z}-module \mathbb{Q}: a subset of M for which every element can be written as an R-linear combination of those generating elements.

Question: For a vector space, its dimension is its minimum number of generators. Can we use the minimum number of generators as a measure of the size of a module in general?

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