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
Below are some equivalent definitions of 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.
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. More generally, a unique factorization domain that is also a one-dimensional domain.For full proof, refer: Unique factorization and one-dimensional iff principal ideal
|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|
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
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.