# Why are Noetherian Rings Special?

/In short, "Noetherian-ness" is a property which generalizes "PID-ness." As Keith Conrad so nicely puts it,

The property of all ideals being singly generated is often not preserved under common ring-theoretic constructions (e.g. $\mathbb{Z}$ is a PID but $\mathbb{Z}[x]$ is not), but the property of all ideals being finitely generated *does* remain valid under many constructions of new rings from old rings. For example... every quadratic ring $\mathbb{Z}[\sqrt{d}]$ is Noetherian, even though many of these rings are not PIDs." (italics added)

*new*ring such that it, too, is Noetherian?" Yep. You can construct the polynomial ring $R[x]$ and it will be Noetherian whenever $R$ is. For more on the Noetherian property, see here.