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:

1. Every ideal in it is principal, viz., it is a principal ideal ring
2. Every prime ideal in it is principal
3. It admits a Dedekind-Hasse norm

Equivalence of definitions

• For the equivalence of definitions (1) and (2), refer Principal ideal ring iff every prime ideal is principal.
• For (1) implies (3), refer principal ideal domain admits Dedekind-Hasse norm.
• For (3) implies (1), refer Dedekind-Hasse norm implies principal ideal ring.

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). For full proof, refer: Noetherian and Bezout iff principal ideal
• A unique factorization domain that is also a Bezout domain. For full proof, refer: Unique factorization and Bezout iff principal ideal
• A unique factorization domain that is also a Dedekind domain. For full proof, refer: Unique factorization and Dedekind iff principal ideal

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
• Normal domain
• Noetherian normal domain
• One-dimensional Noetherian domain
• One-dimensional domain
• Elementary divisor domain
• gcd domain
• Principal ideal ring
• Noetherian ring
• Bezout ring

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
• A unique factorization domain and a Bezout domain: Further information: Unique factorization and Bezout iff principal ideal

Metaproperties

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. For full proof, refer: Polynomial ring over a ring is a PID iff the 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.

Closure under quotients by prime ideals

This property of integral domains is prime-quotient-closed: the quotient of any integral domain satisfying this property by a prime ideal also satisfies the property. Note that we need the ideal to be prime for the quotient to also be an integral domain.
View other prime-quotient-closed properties of integral domains

The quotient of a principal ideal domain by any prime ideal is again a principal ideal domain. In fact, the quotient is either equal to the original domain (in case the prime ideal is zero) or is a field (in case the prime ideal is maximal. This is because in a principal ideal domain, every nonzero prime ideal is maximal.

However, it is true in slightly greater generality that the quotient of a principal ideal ring by any ideal is again a principal ideal ring. For full proof, refer: Principal ideal ring is quotient-closed

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.