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# Principal ideal domains

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

  Bases: `sage.categories.category_singleton.Category_singleton`

  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]

  See also: http://en.wikipedia.org/wiki/Principal_ideal_domain

  TESTS:

      sage: TestSuite(PrincipalIdealDomains()).run()

  class `ElementMethods`
  class `PrincipalIdealDomains.``ParentMethods`
  `PrincipalIdealDomains.``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()

  `PrincipalIdealDomains.``super_categories`()

    EXAMPLES:

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

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