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.
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

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

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.