Periodic rings with commuting nilpotents

Let R be a ring (not necessarily with identity) and let N denote the set of nilpotent elements of R . Suppose that (i) N is commutative, (ii) for every x in R , there exists a positive integer k = k ( x ) and a polynomial f ( λ ) = f x ( λ ) with integer coefficients such that x k = x k + 1 f ( x ) , (iii) the set I n = { x | x n = x } where n is a fixed integer, n > 1 , is an ideal in R . Then R is a subdirect sum of finite fields of at most n elements and a nil commutative ring. This theorem, generalizes the x n = x theorem of Jacobson, and (taking n = 2 ) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume that I n is a subring of R .