Principal ideal domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
The corrresponding general property for commutative unital rings is: principal ideal ring
View other properties of integral domains | View all properties of commutative unital rings
VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

An integral domain is termed a PID or Principal Ideal Domain if it satisfies the following equivalent conditions:

• Every ideal in it is principal, viz., it is a principal ideal ring
• Every prime ideal in it is principal
• It admits a Dedekind-Hasse valuation

Note that the two conditions need not be equivalent when the underlying ring is not a domain.

Relation with other properties

Expression as a conjunction of other properties

• A principal ideal ring that is also an integral domain.
• A Noetherian domain that is also a Bezout domain (equivalently, an integral domain that is both a Noetherian ring and a Bezout ring).
• A unique factorization domain that is also a Dedekind domain.

Stronger properties

• Euclidean domain: For proof of the implication, refer Euclidean implies PID and for proof of its strictness (i.e. the reverse implication being false) refer PID not implies Euclidean
• Polynomial ring over a field

Weaker properties

• Dedekind domain: For proof of the implication, refer PID implies Dedekind and for proof of its strictness (i.e. the reverse implication being false) refer Dedekind not implies PID
• Bezout domain: For proof of the implication, refer PID implies Bezout and for proof of its strictness (i.e. the reverse implication being false) refer Bezout not implies PID
• Noetherian domain
• Unique factorization domain: For proof of the implication, refer PID implies UFD and for proof of its strictness (i.e. the reverse implication being false) refer UFD not implies PID
• Elementary divisor domain

Conjunction expressions

A ring is a principal ideal domain iff it is:

• A principal ideal ring and an integral domain: This is a tautological statement
• A Noetherian ring and a Bezout domain: Further information: Noetherian and Bezout iff principal ideal
• A unique factorization domain and a Dedekind domain: Further information: Unique factorization and Dedekind iff principal ideal

Metaproperties

Polynomial-closedness

This property of commutative unital rings is not closed under passing to the polynomial ring

The polynomial ring over a PID need not be a PID. Two examples are the polynomial ring over the integers, and the polynomial ring in two variables over a field. In fact, the polynomial ring over a ring is a PID iff that ring is a field.

Closure under taking localizations

This property of integral domains is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.
View other localization-closed properties of integral domains | View other localization-closed properties of commutative unital rings

If we localize a principal ideal domain at any multiplicatively closed subset that does not contain zero, and in particular if we localize relative to a prime ideal, we continue to get a principal ideal domain. The reason is, roughly, that any ideal in the localization is generated by the element in the contraction that generates it.

Module theory

Further information: structure theory of modules over PIDs

Any finitely generated module $M$ over a PID $R$ can be expressed as follows:

$M=R/(d_{1})\oplus R/(d_{2})\oplus R/(d_{n})$

where $d_{1}|d_{2}|\ldots |d_{n}$. Some of the $d_{n}$ could be zero.

The $d_{i}$ are unique upto units; the principal ideals they generate are unique.

There is another equivalent formulation:

$M=R/(p_{1}^{k_{1}})\oplus R/(p_{2}^{k_{2}})\oplus R/(p_{r}^{k_{r}})$

Where all the $p_{i}$ are prime.

Thus, a finitely generated module over a PID is projective if and only if it is free.

External links

Definition links

• Wikipedia page
• Planetmath page
• Mathworld page