Derivable maps and derivational points

For an algebra A and an A-bimodule M, let L(A,M) be the set of all linear maps from A to M. A map δ∈L(A,M) is called derivable at C∈A if δ(A)B+Aδ(B)=δ(C), for all A,B∈A with AB=C. We call an element C∈A a derivational point of L(A,M) if the condition δ is derivable at C implies δ is a derivation. We characterize derivable maps by means of Peirce decompositions and determine derivational points for some general bimodules. As a special case, we see that for a nest algebra A on a Hilbert space H, every 0≠C∈A is a derivational point of L(A,B(H)).