A Note on Multiplicative (Generalized)-Derivations and Left Sided Ideals in Semiprime Rings

Let R be a 2-torsion free semiprime ring with center Z(R) and \(\lambda \) a nonzero left sided ideal of R. Let \(F,G:R\rightarrow R\) be multiplicative (generalized)-derivations associated with the map \(d:R\rightarrow R\) (not necessarily additive nor derivation) and \(H,T:R\rightarrow R\) be any two maps. The main goal of this article is to study identities: