AN ITERATION TECHNIQUE AND COMMUTATIVITY OF RINGS

Through much shorter proofs, some new commutativity theorems for rings with unity have been obtained. These results either extend or generalize a few well-known theorems. Our method of proof is based on an iteration technique.

Our objective in this paper is to present some new commutativity theorems for rings with unity using an iteration type technique developed by Tong [4].
Throughout the rest of the paper, R stands for an associative ring with unity I.As usual, [x,y] xy-yx.
The following results will be frequently used in the sequel.
Let R be a ring with unity and f R R be a function such that f(x+l) f(x) for every x in R. If for some n positive interger n, x f(x) =0 for all x in R, then f(x) 0. r LEMMA 1.2.(Tong [4]).Let R be a ring with unity I. Let I r (x) -x o If k>l, let I r r rk_ k(X) I k_l(X+l)-I (x).Then Irr-l(X) F (r-l)r'. + r'x;.Irr(x) =r'., and Irj(x) 0 for J > r.

RESULTS.
A generalization of a famous result due to Bell [6] was obtained by Psomopoulos [I] through a tedious proof.Here we present an entirely different but easier and shorter proof of a similar result.THEOREM 2.1.Let n(>l) and m be two fixed positive integers and t, s be any non- negative integers.
Let R be an associative ring with unity.
Suppose that x [xn,y] [x, Ix s for all x,y in R. If, further, R is m!-torslon free, then R is commmta t ire.
PROOF.Following Tong [4], we set lj(y)= Imj(y), for j=0, 1,2 Ix, I l(y)] x o. (2.3) Further, replacing y by l+y in (2.3) and again using Lemma 1.2, we obtain s Ix, 12 (y)] x O. (2.4) Let y--l+y in (2.4).Then we get  By the hypothesis of the theorem, and the Lemma I.I, the commmtativity of R is obvious.This completes the proof.
COROLLARY I. Let R be a ring with unity and n > I, m be fixed positive integers.
If the identity [xn,y] [x,ym]holds for x,y in R, and further R is m- torsion free, then R is commtatlve.COROLLARY 2. (Psomopoulos et al [7]).Let n m be fixed integers with mn > I, and let R be an s-unital ring.Suppose that every commutator in R is m!torsion free.
If, further, R satisfies the polynomial identity [xn,y] More recently, generalizing a result of Wei Zong Xuan [2], Quadri et al [8] proved that a semi prime Our next result is indeed motivated by the above commutativity conditions.THEOREM 2.2.
Let R be a ring with identity in which any one of the following properties hold for all x,y in R: (PI) Let lj (y) I (y), J 0,1,2 ..... Then by Lemma 1.2 10(Y =yn and we have [xm,y] 10(Y) =y[xm,y] y. ( Replacing y by l+y in the above identity yields [xm, l+y] 10(l+y)= (l+y) [xm, l+y] (l+y).
Setting y l+y in the above identity we get [xm,y] 13(Y) 0.
(2.11) By setting y l+y and iterating (2.11) n times we get [xm,y] In(Y) O.
In view of Lemma 1.2, the above identity gives n! [xm,y] 0.
(2.13) m Now we shall apply iteration to x Put x m 10(x), and then set x l+x in (2.13) which is then reduced to [10(l+x), y] 0.
Using Lemma 1. holds for all x,y, in R. If the characteristic of R does not divide n(m!) (n!), then R is commutative.
PROOF.Let lj(y) Inj(y), for j= 0,I,2,... (1) For re=n, Theorem 2.4 is reduced to Theorem A of Harmancl [3].However, our proof is much shorter and does not use any comblnatorlal type arguments. (ll) As in the remark given after Theorem 2.2, [emma 1.1 can very well be applied to conclude the commutatlvlty of R when it is assumed to be only n(n!)-torslon free.
Thus in this way we get a slight improvement over the result of Harmancl [3].

(2. 5 )
Again, replacing y by l+y in the above identity and then iterating m-I times, we An application of Lemma 1.2 now yields s m! [y, x] x O.
which either (i) [xmy nxy x,y] 0 or xy x,x] 0 holds for all x, y in R, is necessarily commutative.
2 and iterating the above identity m-I times, we get m! Ix,y]