Linear mappings derivable at some nontrivial elements

Suppose that A is an algebra and M is an A-bimodule. Let A be any element in A. A linear mapping δ from A into M is said to be derivable at A if δ(ST)=δ(S)T+Sδ(T) for any S,T in A with ST=A. Given an algebra A, such as a non-abelian von Neumann algebra or an irreducible CDCSL algebra on a Hilbert space H with dimH?2, we show that there exists a nontrivial idempotent P in A such that for any Q∈PAP which is invertible in PAP, every linear mapping derivable at Q from A into some unital A-bimodule (for example, A or B(H)) is derivation.