It seems to me that these 3 algebraic systems are closely related, but it always seems to be Noetherian rings rather than Noetherian domains appearing, and conversely I rarely seem to see principal ideal rings or Dedekind rings.



Why are principal ideal domains and Dedekind domains prominent, but I always seem to see Noetherian rings rather than Noetherian domains?
Dear Negative refraction, 


I would guess that it reflects the particular literature you are looking at. If you were to look at algebraic geometry literature, you would very often see the following line (or a variant thereof)): let $U =$ Spec $A$ be open affine in the irreducible variety $X$. The ring $A$ will then be a Noetherian domain, and Noetherian domains thus arise all the time in algebraic geometry. 


To a certain extent, though (as is more-or-less pointed out in the comments above), the theory of Noetherian domains is the intersection of the theory of Noetherian rings and the theory of domains, i.e. there is not so much more that the general theory says in the case of Noetherian domains specifically that is not a consequence of immediately combining the general theory of Noetherian rings with the assumption that the ring is a domain. So in a commutative algebra book the set of theorems that specifically address Noetherian domains will probably be fairly sparse.


Here is an example of one, though: if $A$ is a Noetherian domain, and $I$ is a non-unit ideal of $A$, then $\cap\_{n \geq 0} I^n = 0$, a form of the Krull intersection theorem which is valid in a Noetherian domain. 


Another point in the general theory where the domain condition arises naturally is in the theory of normal (= integrally closed) rings. A Noetherian ring is defined to be normal if it is a product of finitely many integrally closed domains, and so in the theory of normal Noetherian rings, one frequently reduces to the case of an integrally closed Noetherian domain.


