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