Dhara-Rehman-Raza's identities on left ideals of prime rings

It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the “optimal” case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, $\mathbf{\text{16}}$ (2015), 769 – 779] to the context of nonzero left ideals.