Characterizations of derivations on triangular rings: Additive maps derivable at idempotents

Let T be a triangular ring. An element Z∈T is said to be a full-derivable point of T if every additive map δ from T into itself derivable at Z (i.e. δ(A)B+Aδ(B)=δ(Z) for every A,B∈T with AB=Z) is in fact a derivation. In this paper, under some mild conditions on triangular ring T, we show that some idempotent elements of T are full-derivable points. As an application, we get that, for any nontrivial nest N in a factor von Neumann algebra R, every nonzero idempotent element Q satisfying PQ=Q, QP=P for some projection P∈N is a full-derivable point of the nest subalgebra of R.