Let R be a prime ring with center Z(R), \lambda a nonzero left ideal, \alpha, \beta are automorphisms of R and R admits a generalized (\alpha,\beta)-derivation F associated with a nonzero (\alpha,\beta)-derivation d such that d(Z(R))\neq (0). In the present paper, we prove that if any one of the following holds:
(i) F([x,y])-b\alpha(x\circ y)\in Z(R)
(ii) F([x,y])+b\alpha(x\circ y)\in Z(R)
(iii) F(x \circ y)-b\alpha([x,y])\in Z(R)
(iv) F(x \circ y)+b\alpha([x,y])\in Z(R)
for all x,y\in \lambda and for some b \in R then R is commutative. Also some related results have been obtained.

Additional Information


Rahaman, Md Hamidur