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 (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