Notes on Commutativity of Prime Rings

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