In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.
Contents
- The prehistory of Dedekind domains
- Alternative definitions
- Some examples of Dedekind domains
- Fractional ideals and the class group
- Finitely generated modules over a Dedekind domain
- Locally Dedekind rings
- References
A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID.
The prehistory of Dedekind domains
In the 19th century it became a common technique to gain insight into integral solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer
For a few small values of
By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust. For instance the ring of ordinary integers is a PID, but as seen above the ring
Another illustration of the delicate/robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property—a Noetherian domain
Alternative definitions
For an integral domain
(DD1) Every nonzero proper ideal factors into primes.
(DD2)
(DD3) Every nonzero fractional ideal of
(DD4)
(DD5)
Thus a Dedekind domain is a domain that satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste. In practice, it is often easiest to verify (DD4).
A Krull domain is a higher-dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension 1. This notion can be used to study the various characterizations of a Dedekind domain. In fact, this is the definition of a Dedekind domain used in Bourbaki's "Commutative algebra".
A Dedekind domain can also be characterized in terms of homological algebra: an integral domain is a Dedekind domain if and only if it is a hereditary ring; i.e., every submodule of a projective module over it is projective. Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
Some examples of Dedekind domains
All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.
The ring
The other class of Dedekind rings that is arguably of equal importance comes from geometry: let C be a nonsingular geometrically integral affine algebraic curve over a field k. Then the coordinate ring k[C] of regular functions on C is a Dedekind domain. Indeed, this is essentially an algebraic translation of these geometric terms: the coordinate ring of any affine variety is, by definition, a finitely generated k-algebra, so Noetherian; moreover curve means dimension one and nonsingular implies (and, in dimension one, is equivalent to) normal, which by definition means integrally closed.
Both of these constructions can be viewed as special cases of the following basic result:
Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.
Applying this theorem when R is itself a PID gives us a way of building Dedekind domains out of PIDs. Taking R = Z this construction tells us precisely that rings of integers of number fields are Dedekind domains. Taking R = k[t] gives us the above case of nonsingular affine curves.
Zariski and Samuel were sufficiently taken by this construction to pose as a question whether every Dedekind domain arises in such a fashion, i.e., by starting with a PID and taking the integral closure in a finite degree field extension. A surprisingly simple negative answer was given by L. Claborn.
If the situation is as above but the extension L of K is algebraic of infinite degree, then it is still possible for the integral closure S of R in L to be a Dedekind domain, but it is not guaranteed. For example, take again R = Z, K = Q and now take L to be the field
Fractional ideals and the class group
Let R be an integral domain with fraction field K. A fractional ideal is a nonzero R-submodule I of K for which there exists a nonzero x in K such that
Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums
For any fractional ideal I, one may define the fractional ideal
One then tautologically has
A principal fractional ideal is one of the form
A domain R is a PID if and only if every fractional ideal is principal. In this case, we have Frac(R) = Prin(R) =
For a general domain R, it is meaningful to take the quotient of the monoid Frac(R) of all fractional ideals by the submonoid Prin(R) of principal fractional ideals. However this quotient itself is generally only a monoid. In fact it is easy to see that the class of a fractional ideal I in Frac(R)/Prin(R) is invertible if and only if I itself is invertible.
Now we can appreciate (DD3): in a Dedekind domain—and only in a Dedekind domain! -- is every fractional ideal invertible. Thus these are precisely the class of domains for which Frac(R)/Prin(R) forms a group, the ideal class group Cl(R) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.
We note that for an arbitrary domain one may define the Picard group Pic(R) as the group of invertible fractional ideals Inv(R) modulo the subgroup of principal fractional ideals. For a Dedekind domain this is of course the same as the ideal class group. However, on a more general class of domains—including Noetherian domains and Krull domains—the ideal class group is constructed in a different way, and there is a canonical homomorphism
Pic(R)which is however generally neither injective nor surjective. This is an affine analogue of the distinction between Cartier divisors and Weil divisors on a singular algebraic variety.
A remarkable theorem of L. Claborn (Claborn 1966) asserts that for any abelian group G whatsoever, there exists a Dedekind domain R whose ideal class group is isomorphic to G. Later, C.R. Leedham-Green showed that such an R may constructed as the integral closure of a PID in a quadratic field extension (Leedham-Green 1972). In 1976, M. Rosen showed how to realize any countable abelian group as the class group of a Dedekind domain that is a subring of the rational function field of an elliptic curve, and conjectured that such an "elliptic" construction should be possible for a general abelian group (Rosen 1976). Rosen's conjecture was proven in 2008 by P.L. Clark (Clark 2009).
In contrast, one of the basic theorems in algebraic number theory asserts that the class group of the ring of integers of a number field is finite; its cardinality is called the class number and it is an important and rather mysterious invariant, notwithstanding the hard work of many leading mathematicians from Gauss to the present day.
Finitely generated modules over a Dedekind domain
In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain (PID), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.
Let us briefly recall the structure theory in the case of a finitely generated module
(M1)
differ only in the order of the factors.
(M2) The torsion submodule is a direct summand: i.e., there exists a complementary submodule
(M3PID)
Now let
(M3DD)
if and only if
and
Rank one projective modules can be identified with fractional ideals, and the last condition can be rephrased as
Thus a finitely generated torsionfree module of rank
Theorem: Let R be a Dedekind domain. Then
These results were established by Ernst Steinitz in 1912.
An additional consequence of this structure, which is not implicit in the preceding theorem, is that if the two projective modules over a Dedekind domain have the same class in the Grothendieck group, then they are in fact abstractly isomorphic.
Locally Dedekind rings
There exist integral domains