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

Question

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.

Answer 1

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.

Answer 2

The integral domains are certainly a very nice subclass of the class of all commutative rings. Many classical examples and constructions give rise to domains. One explanation for this is that the theory of commutative rings arose, in part, as a generalization of the theory of fields, and the subrings of fields are precisely the integral domains. Other historically important examples include rings of algebraic or analytic functions, and the necessary and sufficient conditions for these rings to be domains are natural enough to be encountered earlier in the theory of the subject. For instance, the ring of holomorphic functions on an open subset $U \subset \mathbb{C}$ is a domain iff $U$ is what is itself often called a domain in complex function theory: namely, a connected open set.

As for the specific examples of your question: there are some parts of commutative algebra where restricting to domains brings essential simplifications and there are other parts where it doesn't seem to help at all. One of the examples of the latter is ideal theory: the condition that the ideals of a ring be finitely generated is a very natural one. One sign of the naturality is that it is inherited by all quotient rings, not just quotients by prime ideals. (Note though that M. Emerton in his answer has given an example of a result which is particular to Noetherian domains.) In fact there is a theorem that a ring is Noetherian iff all of its prime ideals are finitely generated: this result seems to be inviting us to consider the class of all Noetherian rings rather than just Noetherian domains.

In contrast, the theory of factorization becomes significantly more complicated in (commutative!) rings that are not domains. It is telling that you speak of "Dedekind rings", but in fact in all my experience the term "Dedekind ring" is synonymous with "Dedekind domain": i.e., the concept is only defined for domains. The class of Dedekind domains is amazingly natural in that there are on the order of $100$ nice characterizations of Dedekind domains among all integral domains. If you look at all these conditions on the class of all commutative rings, then they fail to become equivalent, and the graph of implications between them has been the subject of much research by specialists in this area.

I am definitely not an expert on ideal theory in rings which are not domains, so I'm not sure to what extent the research here has converged to standard extensions of the definitions of the usual classes of domains to the case of arbitrary commutative rings. For instance, it seems that it is by now agreed upon that a Prufer ring is a ring in which every finitely generated regular ideal is invertible: see for instance the last chapter of Larsen and McCarty's Multiplicative Ideal Theory. On the other hand, the theory of Prufer rings is definitely more complicated than that of Prufer domains: for instance a local domain is a Prufer ring iff it is a valuation ring, but this does not hold for non-domains.

If I were to take a guess at the "right" definition of a Dedekind non-domain, I guess it would have to be a Noetherian Prufer ring. But I have never seen this definition used for anything. In particular, perhaps it is telling that the standard definition of a Dedekind scheme is a scheme which is integral, normal, Noetherian and of dimension one, so that an affine Dedekind scheme is precisely the spectrum of a Dedekind domain.

As regards principal rings, recall the following theorem of Zariski-Samuel (a proof can be found in the section on principal rings of these notes): any principal ring is a finite direct product of rings $R_i$, each of which is either a PID or a local Artinian principal ring. In particular, if you want a principal ring to be nonsingular in the sense of scheme theory, then it must simply be a finite product of PIDs. Presumably something similar holds for Dedekind non-domains?

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As a sort of afterthought, the following result seems important in the role that integral domains play in arithmetic geometry: a regular local ring is necessarily an integral domain.

Answer 3

Let me offer a somewhat elementary version of what others have written. PID and Dedekind domains were studied because people were interested in factorization properties. In non-domains factorization behaves very badly ($0$ can be factored non-trivially, and that is bad).

To be more precise: the unique factorization property means that for non-zero elements $a,b,c,d$, $ab=cd$ forces $a=c, b=d$ up to units and reordering. In a domain, at least a special case is true: $ab=ac$ implies $b=c$. Without this, it will be very hard to solve equations. That's why (I think) domains are usually assumed.

As for your second question: when study Noetherian rings in general, we simply can't afford to remain in the domain world. Many important classes of rings commutative algebraists think about on a daily basis are naturally not domains: Artinian rings, complete intersections, Cohen-Macaulay, etc.

An example to illustrate my point: the local Grothendieck-Lefschetz theorem says that if $R=S/(f)$ is a local hypersurface which is locally regular in codimension $3$, then $R$ is a UFD! The only proof I know, which was presented very nicely in EGA, uses the Picard groups of the punctured spectra of the rings $R_i = S/(f^i)$. So, in order to study the UFD property of a hypersurface domain, one is forced to introduce non-domains. I hope this example will be as convincing to you as it was to me.

Answer 4

Let me add a somewhat naive and incomplete thought (as opposed to the nice detailed answers of Emerton and Pete).

PIDs and Dedekind domains, even without the domain part are much more specific than Noetherian rings or even domains. The difference is very clear geometrically: The former correspond to smooth curves (or regular $1$-dimensional schemes if you like) while the latter to general noetherian schemes (or reduced irreducible schemes if you add domain).

So, I would argue that when you study something (so) special as curves, it is natural to devote a large amount of your efforts to smooth curves. In contrast, if you are already studying general schemes, then there are advantages to not restrict to reduced and irreducible ones (see some of the comments, especially by BCnrd to Pete's answer). In addition, we obviously also want to study singular curves, even non-reduced ones. However, those are better covered by studying noetherian rings than any PI or Dedekind non-domains you may be able to cook up.