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
Localization of a ring "S" to be multiplicatively closed in what follows. In the case "R" is an integral domain there is an easy construction of the localization. Since the only ring in which 0 is a unit is the trivial ring {0}, the localization "R*" is {0} if 0 is in "S". Otherwise, the field of fractions "K" of "R" can be used: we take "R*" to be the subset of "K" consisting of the elements of the form "r"/"s" with "r" in "R" and "s" in of "S"; as we have supposed "S" multiplicatively closed, "R"* is a subring. The standard embedding