# 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?

I’ve heard it said that the concept of modules was invented to put ideals of rings on the same footing as quotient rings. Both the inclusion of an ideal $I$ into a ring $R$ and the quotient map $R \to R/I$ are homomorphisms of $R$-modules. In fact, these two homomorphisms have a special relationship: the elements of $R$ that are sent to zero in $R/I$ are exactly those in the image of $I\to R$.