  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, \textbf{16}
(2015), 769 -- 779] to the context of nonzero left ideals.
