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
Von Neumann regular ring Von Neumann regular ring In mathematics, a von Neumann regular ring is a ring "R" such that for every "a" in "R" there exists an "x" in "R" such that . To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat. One may think of "x" as a "weak inverse" of "a". In general "x" is not uniquely determined by "a". Von Neumann regular rings were