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 (mathematics) elements in formula_175, and, moreover, any ring homomorphism from "R" that "inverts" "S" uniquely factors through formula_175. The ring formula_175 is called the localization of "R" with respect to "S". For example, if "R" is a commutative ring and "f" an element in "R", then the localization formula_180 consists of elements of the form formula_181 (to be precise, formula_182) The localization is frequently applied to a commutative ring "R" with respect to the complement of a prime ideal (or a union of prime ideals) in "R". In that case formula_183, one often writes formula_184 for formula_175. formula_184 is then a