Download MATH 254A: DEDEKIND DOMAINS I 1. Properties of Dedekind

MATH 254A: DEDEKIND DOMAINS I
BRIAN OSSERMAN
1. Properties of Dedekind domains
We begin by recalling two theorems about Dedekind domains which we plan to
prove.
Theorem 1.1. Every non-zero ideal of a Dedekind domain may be uniquely factored
as a product of prime ideals, up to reordering.
Definition 1.2. Let R be an integral domain with fraction field K. We say that
I ⊂ K is a fractional ideal of R if it is closed under addition and under scalar
multiplication by elements of R, and if there exists a non-zero d ∈ R such that
dI ⊂ R. A fractional ideal is principal if it is of the form αR, for some α ∈ K.
Given fractional ideals I, J of R, the product IJ is defined to be
X
{α ∈ L : α =
iℓ jℓ , iℓ ∈ I, jℓ ∈ J}
ℓ
The product of two fractional ideals is easily seen to be a fractional ideal.
Theorem 1.3. The set of fractional ideals of a Dedekind domain form a group
under multiplication.
We will prove these two theorems next time. However, as a result of the second
theorem, we are able to define:
Definition 1.4. Given a Dedekind domain R, the ideal class group of R is
defined to be the group of fractional ideals modulo the group of principal fractional
ideals.
Thus, the ideal class group measures how far a Dedekind domain is from being
a principal ideal domain. We claim in our context, this is equivalent to measuring
how far away every irreducible element is from being prime. Specifically:
Proposition 1.5. Let R be an integral domain.
(i) If R is Noetherian, then R is a unique factorization domain if and only if
every irreducible element is prime.
(ii) R is a principal ideal domain if and only if R is a Dedekind domain and a
unique factorization domain.
Note that although this is intended as motivation for Theorems 1.1 and 1.3, the
only part which requires either one is the ⇐ direction of (ii); we will use the other
statements in the proofs of the Theorems. To prove the proposition, we will want
to use the following:
Exercise 1.6. Show that a ring R is Noetherian if and only if every ideal is finitely
generated.
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BRIAN OSSERMAN
Proof of Proposition. For (i), we have already observed that a UFD has every irreducible element prime, by definition. Conversely, given an element x ∈ R, we
claim we can write x as a product of irreducibles: if not, it is clear that we can
write x = x1 y, with neither of x1 , y a unit, and where x1 cannot be written as a
product of irreducibles. Repeating this inductively, we obtain a sequence xi with
x0 = x, and xi+1 |xi for each i, with xxi+1
not a unit in R. But then the ideals
i
generated by the xi form an infinite ascending chain, contradicting the hypothesis
that R is Noetherian. Thus, x may be factored into irreducibles, and it is easy to
check inductively that this factorization is unique, using that every irreducible is
prime.
For (ii), it follows from the above exercise that every PID is Noetherian. Next,
recall that in a PID, every irreducible element is prime: if x is irreducible, and m is
any prime ideal containing x, because m is principal it must be equal to (x). This
shows by (i) that any PID is a UFD. But the same argument, if x is a generator
for any non-zero prime ideal, shows that (x) is maximal. We already showed that
any UFD is integrally closed, so every PID is also a Dedekind domain, as desired.
For the converse, note that by Theorem 1.1, it suffices to show that every prime
ideal is principal. But given a non-zero prime ideal p, we must have that p contains
an irreducible element (x), by starting with any non-zero element, factoring it into
irreducibles, and applying the definition of prime ideal inductively. But then this
element is prime, and since every non-zero prime ideal is maximal, and (x) ⊂ p, we
conclude that (x) = p, as desired.
2. Dedekind domains and DVRs
There are several proofs of Theorems 1.1 and 1.3. The most classical approach
only works for rings of integers, and first proves that the ideal class group is finite,
and concludes these theorems. See [1, §12.2]. A slightly less direct, but more
general proof is given in [2, §1.6]. We will take a more technology-heavy approach
of proving these theorems via study of local rings. This is a bit longer, but has the
advantage of introducing local rings and the concept of constructing global data
from local data.
We now explore the properties of local rings of Dedekind domains. Recall the
following definitions:
Definition 2.1. Let R be an integral domain with field of fractions K, and S a
multiplicatively closed subset not containing 0. We define S −1 R ⊂ K to be the
subring of the form { sr : r ∈ R, s ∈ S}. If p is a prime ideal of R, we define Rp , the
local ring of R at p, to be S −1 R with S = R r p.
Exercise 2.2. If one considers the map from ideals of Rp to ideals of R given by
I 7→ I ∩ R, this map is injective, and if restricted to prime ideals, gives a bijection
between prime ideals of Rp and prime ideals of R contained in p.
In particular, if R is Noetherian, then Rp is Noetherian.
Definition 2.3. An integral domain R′ is said to be a discrete valuation ring
or DVR if it is a principal ideal domain with a unique maximal ideal.
We have already seen in Proposition 1.5 that every PID is Dedekind, so in
particular every DVR is Dedekind. It follows that the only prime ideals of a DVR
are (0) and the maximal ideal.
We will need the following facts:
MATH 254A: DEDEKIND DOMAINS I
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Exercise 2.4. Show that if R is a Noetherian ring, and I an ideal, such that there
exists a unique prime ideal p containing I, then I contains some power of p.
We next prove two converses to the statement that a DVR is Dedekind.
Lemma 2.5. Any Dedekind domain R′ with a unique maximal ideal is a discrete
valuation ring. Every non-zero ideal of a discrete valuation ring is a power of the
maximal ideal.
Proof. From the definitions, it suffices to check that R′ is a PID. We first claim that
the maximal ideal m is principal: choose a ∈ m non-zero; by the above exercise,
∃n ∈ N such that mn ⊂ (a), but mn−1 6∈ (a). Choose b ∈ mn−1 r (a), and consider
x = ab ∈ K, the field of fractions of R′ . From the construction, we see that x−1 6∈ R′ ,
but x−1 m ⊂ R′ . Now, since R′ is integrally closed, we find that x−1 is not integral
over R′ , so since m is finitely generated, x−1 m 6⊂ m, by the lemma from lecture 2.
But x−1 m ⊂ R′ means it is an ideal of R′ , so if it is not contained in m, it must be
equal to R′ , and we conclude that m = (x).
We next show that m = (x) implies that every ideal is principal. We first claim
that every irreducible element is prime, and more precisely, of the form xu for some
unit u. But given y ∈ R′ irreducible, because y is not a unit, y ⊂ m, so x|y, and by
the definition of irreducibility, y = xu for some unit u, as desired. By Proposition
1.5 (i), it follows that R′ is a UFD, and we see that every element of R′ may be
written as xn u for some n ∈ N ∪ {0}, u a unit. If I is a non-zero ideal of R′ , let
′
n ∈ N ∪ {0} be mina∈I {n′ : a = xn u}; it is then clear that I = (x)n . Thus R′ is a
DVR, as desired.
We have further shown that in R′ , every non-zero ideal is a power of the maximal
ideal, and since we already saw that every DVR is Dedekind, we conclude that this
holds in every DVR.
We will start next time by proving the following stronger statement:
Proposition 2.6. Let R be a Noetherian integral domain which is not a field. Then
R is a Dedekind domain if and only if for all non-zero prime ideals p, the local ring
Rp is a discrete valuation ring.
References
1. Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, second
ed., Springer-Verlag, 1990.
2. Serge Lang, Algebraic number theory, second ed., Springer-Verlag, 1994.