Bulletin of the
Korean Mathematical Society BKMS

Let $\mathcal{R}$ be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map $\delta$ from $\mathcal{R}$ into itself is called a Jordan derivable map at commutative zero point if $\delta(AB+BA)=\delta(A)B+B\delta(A)+A\delta(B)+\delta(B)A$ for all $A, B\in\mathcal{R}$ with $AB=BA=0$. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form $\delta(A)=\psi(A)+CA$ for all $A\in\mathcal{R}$, where $\psi$ is an additive Jordan derivation of $\mathcal{R}$ and $C$ is a central element of $\mathcal{R}$. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.

Keywords: Derivations, Jordan derivable maps, Jordan higher derivable maps, commutative zero points

MSC numbers: Primary 16W25; Secondary 47B47

Supported by: This work is Supported by National Natural Science Foundation of China(No. 11671078)