Ecclesiastical ring the new pope. The pope's ring is not to be confused with the papal seal, which is what the pope uses to seal documents. The Fisherman's Ring is no longer used to seal anything. Except Pope Benedict XVI, who chose to wear his at all times, the ring had not been seen being publicly worn by a pope since the papacy of Pope Pius XII, who refrained from wearing it once his election ceremonies had been concluded. A collection of rings belonging to the popes of the past is exhibited in a glass case in a room within the papal
Ring of integers integers" because of this. The ring of integers of an algebraic number field is the unique maximal order in the field. The ring of integers is a finitely-generated -module. Indeed, it is a free -module, and thus has an integral basis, that is a basis of the -vector space such that each element in can be uniquely represented as with . The rank of as a free -module is equal to the degree of over . The rings of integers in number fields are Dedekind domains. If is a prime, ζ is a th root of unity and is the