Dedekind domain

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.
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 integer solutions of polynomial equations using rings of algebraic numbers of higher degree. For instance, fix a positive integer formula_1. In the attempt to determine which integers are represented by the quadratic form formula_2, it is natural to factor the quadratic form into formula_3, the factorization taking place in the ring of integers of the quadratic field formula_4. Similarly, for a positive integer formula_5 the polynomial formula_6 (which is relevant for solving the Fermat equation formula_7) can be factored over the ring formula_8, where formula_9 is a primitive "n"-th root of unity.
For a few small values of formula_1 and formula_5 these rings of algebraic integers are PIDs, and this can be seen as an explanation of the classical successes of Fermat (formula_12) and Euler (formula_13). By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field formula_14 is a PID was well known to the quadratic form theorists. Especially, Gauss had looked at the case of imaginary quadratic fields: he found exactly nine values of formula_15 for which the ring of integers is a PID and conjectured that there were no further values. (Gauss' conjecture was proven more than one hundred years later by Kurt Heegner, Alan Baker and Harold Stark.) However, this was understood (only) in the language of equivalence classes of quadratic forms, so that in particular the analogy between quadratic forms and the Fermat equation seems not to have been perceived. In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all formula_16; that is, that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring formula_8 is a UFD. Ernst Kummer had shown three years before that this was not the case already for formula_18 (the full, finite list of values for which formula_8 is a UFD is now known). At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents formula_5 using what we now recognize as the fact that the ring formula_8 is a Dedekind domain. In fact Kummer worked not with ideals but with "ideal numbers", and the modern definition of an ideal was given by Dedekind.
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 formula_22 of algebraic integers in a number field formula_23 need not be a PID. In fact, although Gauss also conjectured that there are infinitely many primes formula_24 such that the ring of integers of formula_25 is a PID, we do not even know whether there are infinitely many number fields formula_23 (of arbitrary degree) such that formula_22 is a PID! On the other hand, the ring of integers in a number field is always a Dedekind domain.
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 formula_28 is Dedekind iff for every maximal ideal formula_29 of formula_28 the localization formula_31 is a Dedekind ring. But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring (DVR), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.
Alternative definitions.
For an integral domain formula_28 that is not a field, all of the following conditions are equivalent:
Thus a Dedekind domain is a domain that either is a field, or 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; that is, 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 formula_46 of algebraic integers in a number field "K" is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal "I" of "R", "R"/"I" is a finite set, and recall that a finite integral domain is a field; so by (DD4) "R" is a Dedekind domain. As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples.
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. This is largely clear simply from translating geometric terms into algebra: the coordinate ring of any affine variety is, by definition, a finitely generated "k"-algebra, hence 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 says precisely that rings of integers of number fields are Dedekind domains. Taking "R" = "k"["t"], one obtains the above case of nonsingular affine curves as branched coverings of the affine line.
Zariski and Samuel were sufficiently taken with this construction to ask whether every Dedekind domain arises from it; that is, 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 formula_47 of all algebraic numbers. The integral closure is nothing else than the ring formula_48 of all algebraic integers. Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that formula_48 is not even Noetherian! In general, the integral closure of a Dedekind domain in an infinite algebraic extension is a Prüfer domain; it turns out that the ring of algebraic integers is slightly more special than this: it is a Bézout domain.
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 formula_50
Given two fractional ideals "I" and "J", one defines their product "IJ" as the set of all finite sums formula_51: the product "IJ" is again a fractional ideal. The set Frac("R") of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal "R".
For any fractional ideal "I", one may define the fractional ideal
One then tautologically has formula_53. In fact one has equality if and only if "I", as an element of the monoid of Frac("R"), is invertible. In other words, if "I" has any inverse, then the inverse must be formula_54.
A principal fractional ideal is one of the form formula_55 for some nonzero "x" in "K". Note that each principal fractional ideal is invertible, the inverse of formula_55 being simply formula_57. We denote the subgroup of principal fractional ideals by Prin("R").
A domain "R" is a PID if and only if every fractional ideal is principal. In this case, we have Frac("R") = Prin("R") = formula_58, since two principal fractional ideals formula_55 and formula_60 are equal iff formula_61 is a unit in "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) every fractional ideal is 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
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 formula_29 over a PID formula_28. We define the torsion submodule formula_64 to be the set of elements formula_1 of formula_29 such that formula_67 for some nonzero formula_68 in formula_28. Then:
(M1) formula_64 can be decomposed into a direct sum of cyclic torsion modules, each of the form formula_71 for some nonzero ideal formula_37 of formula_28. By the Chinese Remainder Theorem, each formula_71 can further be decomposed into a direct sum of submodules of the form formula_75, where formula_76 is a power of a prime ideal. This decomposition need not be unique, but any two decompositions
differ only in the order of the factors.
(M2) The torsion submodule is a direct summand. That is, there exists a complementary submodule formula_78 of formula_29 such that formula_80.
(M3PID) formula_78 isomorphic to formula_82 for a uniquely determined non-negative integer formula_5. In particular, formula_78 is a finitely generated free module.
Now let formula_29 be a finitely generated module over an arbitrary Dedekind domain formula_28. Then (M1) and (M2) hold verbatim. However, it follows from (M3PID) that a finitely generated torsionfree module formula_78 over a PID is free. In particular, it asserts that all fractional ideals are principal, a statement that is false whenever formula_28 is not a PID. In other words, the nontriviality of the class group Cl(R) causes (M3PID) to fail. Remarkably, the additional structure in torsionfree finitely generated modules over an arbitrary Dedekind domain is precisely controlled by the class group, as we now explain. Over an arbitrary Dedekind domain one has
(M3DD) formula_78 is isomorphic to a direct sum of rank one projective modules: formula_90. Moreover, for any rank one projective modules formula_91, one has
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 formula_96 can be expressed as formula_97, where formula_37 is a rank one projective module. The Steinitz class for "P" over "R" is the class formula_99 of formula_37 in Cl(R): it is uniquely determined. A consequence of this is:
Theorem: Let "R" be a Dedekind domain. Then formula_101, where K0("R") is the Grothendieck group of the commutative monoid of finitely generated projective "R" modules.
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 formula_28 that are locally but not globally Dedekind: the localization of formula_28 at each maximal ideal is a Dedekind ring (equivalently, a DVR) but formula_28 itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called "proper almost Dedekind rings."