Throughout , R will represent a ring with center C, N the set of nilpotents of R, and E the set of idempotents of R. For each integer n > 1, we set En = {x E R I xn = x}. An element x of R is called potent if x C P = Un=2 En. A ring R is called periodic if for every x G R, x m = x n for some distinct positive integers m, n. By a theorem of Chacron (see [3, Theorem 1]), R is periodic if and only if for each x E R, there exists a positive integer k = k(x) and a polynomial f(A) = fx(A) with integer coefficients such that x k = xk+l f (x ) . The ring R is called weakly periodic if every element of R is expressible as a sum of a ni lpotent element and a potent element of R • R = N + P. It is well-known tha t if R is periodic then it is weakly periodic (see [2]). Whether a weakly periodic ring is necessarily periodic is apparent ly not known. For x, y E R, Ix, Y]i = xy yx is the usual commuta tor , and for every positive integer k > 1, we define inductively [x, Y]k = [[x, Y]k-1, Y]" We also write [x,y] to denote [x,Y]l. For any x , y E R, a word ( m o n o m i a l ) w ( x , y ) i s a product in which each factor is x or y. The empty word is defined to be "1." Finally, the commutator ideal of R will be denoted by C(R) . The major purpose of this paper is to prove the following two theorems.