Ecclesiastical ring gold, stone-set ring. Roman Catholic bishops traditionally have their episcopal ring set with an amethyst. Aside from the rings a bishop purchases or is given by others, his rings belong to the Church; he will have inherited the previous bishop's ring collection, which is held in trust. While all hierarchs are accorded the honor of being buried wearing a ring, all rings belonging to the Church will be returned to the Church upon the retirement or death of any hierarch. In a Decree of Pope Boniface IV (A.D. 610) it describes monks raised to the episcopal dignity as "anulo pontificali
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