Let A be a unital algebra and M be a unital A-bimodule. A characterization of generalized derivations and generalized Jordan derivations from A into M, through zero products or zero Jordan products, is given. Suppose that M is a unital left A-module. It is investigated when a linear mapping from A into M is a Jordan left derivation under certain conditions. It is also studied whether an algebra...

We prove that δ-derivations of a simple finite-dimensional Lie algebra over field characteristic zero, with values in module, are either inner derivations, or, the case adjoint multiplications by scalar, or some exceptional cases related to sl(2). This can be viewed as an extension classical first Whitehead Lemma.

the aim of this paper is to show that under some mild conditions a functional equation of multiplicative (; )-derivation is superstable on standard operator algebras. furthermore, we prove that this generalized derivation can be a continuous and an inner (; )- derivation.