# 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 ringsVIEW 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

A commutative unital ring is termed a **principal ideal domain** (also known as **PID**) if it is an integral domain and a principal ideal ring. Explicitly, it must satisfy the following **equivalent** conditions:

No. | Shorthand | A commutative unital ring is termed a principal ideal domain if ... | A commutative unital ring is termed a principal ideal domain if ... |
---|---|---|---|

1 | every ideal principal | it is an integral domain and every ideal in it is principal | is an integral domain (i.e., implies or ) and for every ideal , there exists such that , i.e., . |

2 | every prime ideal principal | it is an integral domain and every prime ideal in it is principal | is an integral domain (i.e., implies or ) and for every prime ideal , there exists such that , i.e., . |

3 | Dedekind-Hasse norm | it admits a Dedekind-Hasse norm | there is a Dedekind-Hasse norm on . |

### 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 multiplicative Dedekind-Hasse norm.
- For (3) implies (1), refer Dedekind-Hasse norm implies principal ideal ring.

## Examples

Since Euclidean implies PID, all the typical examples of Eulidean domains are also examples of principal ideal domains. There are some PIDs that are not Euclidean domains -- in particular, the domain is not a Euclidean domain because it does not have a universal side divisor.

## Relation with other properties

### Expression as a conjunction of other properties

First component of conjunction | Second component of conjunction | Proof |
---|---|---|

principal ideal ring | integral domain | by definition |

Noetherian domain | Bezout domain | Noetherian and Bezout iff principal ideal |

unique factorization domain | Bezout domain | Unique factorization and Bezout iff principal ideal |

unique factorization domain | Dedekind domain | (see next) |

unique factorization domain | one-dimensional domain | Unique factorization and one-dimensional iff principal ideal |

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Euclidean domain | has a Euclidean norm | Euclidean implies PID | PID not implies Euclidean | click here |

Polynomial ring over a field | for a field | (via Euclidean domain) | (via Euclidean domain) | click here |

### Weaker properties

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

polynomial-closed property of commutative unital rings | No | follows from Polynomial ring over a ring is a PID iff the ring is a field | It is possible to have a principal ideal domain such that the polynomial ring is not a principal ideal domain. In fact, is a principal ideal domain if and only if is a field, so for instance or give counterexamples. |

### 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 over a PID can be expressed as follows:

where . Some of the could be zero.

The are unique upto units; the principal ideals they generate are unique.

There is another equivalent formulation:

Where all the are prime.

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