During another attempt to come to grips with Hillman's excellent book Algebraic Invariants of Links, I am having difficulty figuring out why Krull rings are the setting for Chapter 3- the natural setting for determinantal invariants, such as the Alexander polynomial and Franz-Reidemeister torsion. What is an example of a non-noetherian or non-factorial Krull ring which might conceivably be useful in low-dimensional topology?

In fact, I have a simpler, more general question:

What are examples of non-noetherian Krull rings, or non-factorial Krull rings, which are actually used in mathematics?

In particular, how does the condition for a ring to be Krull manifest itself in terms of the geometry or topology of something, as opposed to some other class such as Grothendieck's "excellent rings"? In another direction, does something good happen for K-theory over Krull rings? Are Krull rings somehow natural objects to take determinants in?