equivalent characterizations of Dedekind domains

equivalent characterizations of Dedekind domains

Dedekind domains

can be defined as

integrally closed

Noetherian

(

http://planetmath.org/Noetherian

) domains in which every nonzero

prime ideal

is

maximal

. However, there are several alternative

characterizations

which can be used, and are listed in the following

theorem

.

Theorem

.

Let

R

be an

integral domain

. Then, the following are

equivalent

.

1.

R

is Noetherian, integrally closed, and every prime ideal is maximal.

2.

Every nonzero

proper ideal

is a

product

of

maximal ideals

.

3.

Every nonzero proper ideal is product of prime ideals.

4.

Every nonzero ideal is

invertible

(

http://planetmath.org/FractionalIdeal

).

5.

Every ideal is projective (

http://planetmath.org/ProjectiveModule

) as an

R

-module.

6.

R

is Noetherian and every

finitely generated

torsion-free

R

-module is projective.

7.

R

is Noetherian and the

localization

R

𝔪

is a

principal ideal domain

for each maximal ideal

𝔪

.

Furthermore, if these properties are satisfied then the decomposition into primes in (

2

) and (

3

) is unique up to reordering of the factors.

For example, if

R

is a principal ideal domain then ideals are clearly invertible and, by condition

4

it is Dedekind, so is integrally closed. In this case, factorization of ideals coincides with

prime factorization

in the ring. For the equivalence of

4

and

1

see proof that a domain is Dedekind if its ideals are invertible.

Once it is known that a ring is Dedekind then conditions

2

and

3

show that we get

unique factorization

of ideals in terms of prime or, equivalently, maximal ideals and

conversely

, Dedekind domains are are the only integral domains in which such decompositions exist (see proof that a domain is Dedekind if its ideals are products of maximals and proof that a domain is Dedekind if its ideals are products of primes).

The equivalence of

4

and

5

is immediate once it is known that

invertible ideals are projective

. For rings which are not integral domains, the property that ideals are projective still makes sense. Such rings are called

hereditary

(

http://planetmath.org/HereditaryRing

), and give one possible

generalization

of the

concept

of Dedekind domains.

As ideals in a Noetherian domain are finitely generated torsion-free

submodules

of

R

, condition

6

clearly implies

5

. Domains which are not necessarily Noetherian, but for which every finitely generated torsion-free module is projective are known as Prüfer domains. The equivalence of

6

and

5

then follows from the alternative characterization of Prüfer domains as integral domains in which every finitely generated ideal is projective.

Condition

7

(see proof that a Noetherian domain is Dedekind if it is locally a PID) shows that for

Noetherian rings

, being a Dedekind domain is a local property (

http://planetmath.org/Localization

) and therefore the notion generalizes to apply to algebraic varieties (

http://planetmath.org/

Variety

) and schemes (

http://planetmath.org/Scheme

).

References

1

P.M. Cohn,

Algebra

. Vol 2

, Second edition. John Wiley & Sons Ltd, 1989.

Title

equivalent characterizations of Dedekind domains

Canonical name

EquivalentCharacterizationsOfDedekindDomains

Date of creation

2013-03-22 18:34:27

Last modified on

2013-03-22 18:34:27

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

10

Author

gel (22282)

Entry type

Theorem

Classification

msc 13A15

Classification

msc 13F05

Related topic

DedekindDomain

Related topic

FinitelyGeneratedTorsionFreeModulesOverPruferDomains

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