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
Graded ring The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to a non-associative algebra as well; e.g., one can consider a graded Lie algebra. Let be a graded ring. Elements of any factor formula_4 of the decomposition are called homogeneous elements of degree "n". Every element "a" of "R" may be uniquely written as a sum "a" = "a" + "a" + ... + "a" with all "a" homogeneous elements of distinct "R". These "a" are called the homogeneous components of "a". Some basic properties are: An ideal