STRONG COMMUTATIVITY PRESERVING GENERALIZED DERIVATIONS ON TRIANGULAR RINGS

Let U = Tri(A, M, B) be a triangular ring such that either A or B has no nonzero central ideals. It is shown that every pair of strong commutativity preserving generalized derivations g(1), g(2) of U (i.e., [g(1)(x), g(2)(y)] = [x, y] for all x, y is an element of U) is of the form g(1)(x) = lambda(-1) x + [x, u] and g(2)(x) = lambda(2)g(1)(x), where lambda is an element of Z(U), the center of U, and u is an element of U with u[U, U] = 0 = [U, U]u. As consequences, every pair of strong commutativity preserving generalized derivations on upper triangular matrix rings and nest algebras is determined.