Existence, automatic continuity and invariant submodules of generalized derivations on modules

Let ${mathcal{A}}$ be a ${mathbb{C}}$ -algebra, δ be a derivation on ${mathcal{A}}$ and ${mathcal{M}}$ be a left ${mathcal{A}}$ -module. A linear map ${tau : mathcal{M} rightarrow mathcal{M}}$ is called a generalized derivation relative to δ if ${tau(am)=atau(m)+delta(a)m,(a in mathcal{A}, m in mathcal{M})}$ . In this article first we study the existence of generalized derivations. In particular we show that free modules and projective modules always have nontrivial generalized derivations relative to nonzero derivations of ${mathcal{A}}$ . Then we investigate the invariance of prime submodules under generalized derivations. Specifically we show that every minimal prime submodule of ${mathcal{M}}$ is invariant under every generalized derivation. Moreover we obtain analogs of Posner’s theorem for generalized derivations. In the case that ${mathcal{A}}$ is a Banach algebra and ${mathcal{M}}$ is a Banach left ${mathcal{A}}$ -module, we study the existence of continuous generalized derivations and automatic continuity of generalized derivations.