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.
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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

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