Abstract

The main theorem proved in the present paper states as follows “Let m, k, n and s be fixed non-negative integers such that k and n are not simultaneously equal to 1 and R be a left (resp right) s-unital ring satisfying [(xmyk)n−xsy,x]=0 (resp [(xmyk)n−yxs,x]=0) Then R is commutative.” Further commutativity of left s-unital rings satisfying the condition xt[xm,y]−yr[x,f(y)]xs=0 where f(t)∈t2Z[t] and m>0,t,r and s are fixed non-negative integers, has been investigated Finally, we extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements x and y for their values. These results generalize a number of commutativity theorems established recently.