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
Commutative ring field) any ideal (such as the one generated by 6) decomposes uniquely as a product of prime ideals. Any maximal ideal is a prime ideal or, more briefly, is prime. Moreover, an ideal "I" is prime if and only if the factor ring "R" / "I" is an integral domain. Proving that an ideal is prime, or equivalently that a ring has no zero-divisors can be very difficult. Yet another way of expressing the same is to say that the complement "R" \ "p" is multiplicatively closed. The localisation ("R" \ "p")"R" is important enough to have its own notation: