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

Let T be a triangular ring. An element Z is an element of T is said to be a full-derivable point of T if every additive map from T into itself derivable at Z (i.e. delta(A)B +A delta(B) = delta(Z) for every A, B is an element of 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 non-trivial nest N in a factor von Neumann algebra R, every nonzero idempotent element Q satisfying PQ = Q. QP = P for some projection P is an element of N is a full-derivable point of the nest subalgebra Alg N of R. (c) 2009 Elsevier Inc. All rights reserved.