Principal ideal domains¶

class sage.categories.principal_ideal_domains.PrincipalIdealDomains(s=None)

The category of (constructive) principal ideal domains

By constructive, we mean that a single generator can be constructively found for any ideal given by a finite set of generators. Note that this constructive definition only implies that finitely generated ideals are principal. It is not clear what we would mean by an infinitely generated ideal.

EXAMPLES:

sage: PrincipalIdealDomains()
Category of principal ideal domains
sage: PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]

class ElementMethods

Bases: object

class ParentMethods

Bases: object

additional_structure()

Return None.

Indeed, the category of principal ideal domains defines no additional structure: a ring morphism between two principal ideal domains is a principal ideal domain morphism.

EXAMPLES:

sage: PrincipalIdealDomains().additional_structure()

super_categories()

EXAMPLES:

sage: PrincipalIdealDomains().super_categories()
[Category of unique factorization domains]