Some conditions for the commutativity of rings

Abstract

Throughout, R will represent an associative ring (may be without unity 1) with Z(R), N ( R ) and C(i t) denoting its centre, the set of nilpotent elements and the commutator ideal of R respectively. For any a, b E R as usual [a, b] = ab ba. In his paper [2] Bell proved that a ring R generated by n th power of its elements and satisfying the polynomial identity [x", y] = [x, yn] for all x, y in R and a fixed positive integer n > 1, is commutative. Motivated by the above observation ttarmanci [4] proved that a ring R with unity 1 satisfying [xk, y] = [x,y k] (k = n , n + 1), is commutative. Recently, Gupta [3] generalized the above result as follows: