4.6 A Characterization of Dedekind Domains

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In this final section we’ll prove the converse of Theorem 4.4.6, thereby

giving a characterization of Dedekind domains.

To begin with, let R be an arbitrary integral domain, with fraction field

E. In analogy with the preceeding section, if I _ R is an ideal, we set

I−1 = {_ 2 E| _I _ R}.

We say that I is invertible if I−1I = R.

Lemma 4.6.1 Assume that I _ R and admits factorizations

P1P2 · · · Pr = I = Q1Q2 · · ·Qs,

1See L. Carlitz, A characterization of algebraic number fields with class number two,

Proc. Amer. Math. Soc. 11 (1960), 391-392. In case R is the ring of integers in a finite

extension of the rational field, Carlitz also proves the converse.

102 CHAPTER 4. DEDEKIND DOMAINS

where the Pi’s and the Qj ’s are invertible prime ideals. Then r = s, and

(possibly after re-indexing) Pi = Qi, i = 1, 2, · · · , r.

Lemma 4.6.2 Let R be an integral domain.

(i) Any non-zero principal ideal is invertible.

(ii) If 0 6= x 2 R, and if the principal ideal (x) factors into prime ideals as

(x) = P1P2 · · · Pr, then each Pi is invertible.

Now assume that R is an integral domain satisying the following condition:

(*) If I _ R is an ideal of R, then there exist prime ideals P1, P2, · · · , Pr _

R such that

I = P1P2 · · · Pr.

Note that no assumption is made regarding the uniqueness of the above

factorization. We shall show that uniqueness automatically follows. (See

Corollary 4.6.9.2 , below.) Of course, this is exactly analogous with what

happens in unique factorization domains.

Our goal is to show that R is a Dedekind domain.

Proposition 4.6.3 Any invertible prime ideal of R is maximal.

Proposition 4.6.4 Any prime ideal is invertible, hence maximal.

Corollary 4.6.4.1 Any ideal is invertible.

Corollary 4.6.4.2 Any ideal of R factors uniquely into prime ideals.

Proposition 4.6.5 R is Noetherian.

Our task of showing that R is a Dedekind domain will be complete as

soon as we can show that R is integrally closed. To do this it is convenient

to introduct certain “overrings” of R, described as below.

Let R be an arbitrary integral domain and let E = F(R). If P _ R is a

prime ideal of R we set

RP = {_/_ 2 E| _, _ 2 R, _ 62 P}.

It should be clear (using the fact that P is a prime ideal) that RP is a

subring of E containing R. It should also be clear that F(RP ) = E. RP is

called the localization of R at the prime ideal P.

4.6. A CHARACTERIZATION OF DEDEKIND DOMAINS 103

Lemma 4.6.6 Let I be an ideal of R, and let P be a prime ideal of R.

(i) If I 6_ P then RP I = RP .

(ii) RPP−1 properly contains RP .

Lemma 4.6.7 If _ 2 E then either _ 2 RP or _−1 2 RP .

The following is now really quite trivial.

Lemma 4.6.8 RP is integrally closed.

Proposition 4.6.9 R = \RP , the intersection taken over all prime ideals

P _ R.

As an immediate result, we get

Corollary 4.6.9.1 R is integrally closed.

Combining all of the above we get the desired characterization of Dedekind

domains:

Corollary 4.6.9.2 R is a Dedekind domain if and only if every ideal of R

can be factored into prime ideals.

Exercises

1. A valuation ring is an integral domain R such that if I and J are ideals

of R, then either I _ J or J _ I. Prove that for an integral domain

R, the following three conditions are equivalent:

(i) R is a valuation ring.

(ii) if a, b 2 R, then either (a) _ (b) or (b) _ (a).

(iii) If _ 2 E := F(R), then either _ 2 R or _−1 2 R.

(Thus, we see that the rings RP , defined above, are valuation rings.)

2. Let R be a Noetherian valuation ring.

(i) Prove that R is a p.i.d.

104 CHAPTER 4. DEDEKIND DOMAINS

(ii) Prove that R contains a unique maximal ideal. (This is true even

if R isn’t Noetherian.)

(iii) Conclude that, up to units, R contains a unique prime element.

(A ring satisfying the above is often called a discrete valuation ring .)

3. Let R be a discrete valuation ring, as in Exercise 2, above, and let

_ be the prime, unique up to associates. Define _(a) = r, where

a = _rb, _ / b. Prove that _ is an algorithm for R, giving R the

structure of a Euclidean domain.

4. Let R be a Noetherian domain and let P be a prime ideal. Show that

the localization RP is Noetherian.

5. Let R be a ring in which every ideal I _ R is invertible. Prove that R is

a Dedekind domain. (Hint: First, as in the proof of Proposition 4.6.5,

R is Noetherian. Now let C be the set of all ideals that are not products

of prime ideals. Since R is Noetherian, C 6= ; implies that C has a

maximal member J. Let J _ P, where P is a maximal ideal. Clearly

J 6= P. Then JP−1 _ PP−1 = R and so JP−1 is an ideal of R;

clearly J _ JP−1. If J = JP−1, then JP−1 = P1P2 · · · Pr so J =

PP1P2 · · · Pr. Thus J = JP−1 so JP = J. This is a contradition,

why?)

6. Here is an example of a non-invertible ideal in an integral domain R.

Let

R = {a + 3b

p

−5| a, b 2 Z},

and let I = (3, 3

p

−5), i.e., I is the ideal generated by 3 and 3

p

−5.

Show that I is not invertible. (An easy way to do this is to let J = (3),

the principal ideal generated by 3, and observe that despite the fact

that I 6= J, we have I2 = IJ.)

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