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
Partially ordered ring a ring "A" to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists. If "S" is a subset of a ring "A", and: then the relation formula_1 where formula_2 iff formula_24 defines a compatible partial order on "A" ("ie." formula_8 is a partially ordered ring). In any l-ring, the "absolute value" formula_26 of an element "x" can be defined to be formula_27, where formula_28 denotes the maximal element. For any "x" and "y", holds.