Prüfer domains: searching for an intuitive approach

Question

Prüfer domains are domains in which every finitely generated ideal is invertible. There are many definitions, like all localizations in prime ideals of a Prüfer domain is a valuation domain. But I'm searching for a more intuitive explanation for these domains in order to understand their structure better.

Answer 1

I like to think of them as non-Noetherian Dedekind domains. A lot of the properties that hold for the category of modules over a Dedekind domains hold for the category of finitely generated modules over a Prüfer domain.

Lattice theoretically, they are integral domains whose lattice of ideals is distributive.

Homologically, all their finitely generated ideals are projective.

Kaplansky also characterized Prüfer domains as the domains in which the torsion submodule of every finitely generated module splits out.

Perhaps you should just go chase the extensive lists of equivalent conditions. Equivalences do more than anything to teach you what a particular condition is like, and Prüfer domains have a TON of characterizations.