Abstract
Let R be an associative ring with unity. It is proved that if R satisfies the polynomial identity [xny − ymxn, x] = 0(m > 1, n ≥ 1), then R is commutative. Two or more related results are also obtained.
Highlights
Throughout this paper, R will be an associative ring, Z(R) the center of R, N the, set of all nilpotent elements of R, N’ the set of all zero divisors of R, and C(R) the commutator ideal of R
For any pair of elements y in R, we set as usual Ix,y] xy yx
In [4], Psomopoulos has shown that an s-unital ring R in which the polynomial identity [xny y x,x] 0 (m > I, n > 0) holds, must be commutative
Summary
Throughout this paper, R will be an associative ring, Z(R) the center of R, N the. , set of all nilpotent elements of R, N’ the set of all zero divisors of R, and C(R) the commutator ideal of R. Generalizing some results from Bell [1] Quadrl and Khan [2,3]) proved ym m, that if R is a ring satisfying the polynomial identity [xy x x] 0. In [4], Psomopoulos has shown that an s-unital ring R in which the polynomial identity [xny y x,x] 0 (m > I, n > 0) holds, must be commutative. In this paper, motivated by the above polynomial identities, we intend to prove results on commutativlty of a ring R with unity satisfying the fo]lowin property:. Positive Integer k, ix y] k xk-l[x,y]. Suppose that for some positive integer k, k x y k(x+|) y for all x, y
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