In this paper we study
properties of associative derivations whose iterates are related in rather special ways
to the original derivation, or to the iterates of another derivation. An associative
derivation d : R → R is an additive (or linear when appropriate) mapping on a ring R
satisfying d(xy) = xd(y) + d(x)y for all x,y ∈ R. A derivation d : R → R is called
inner if d(x) = (ada)(x) for some a ∈ R where (ada)(x) = [a,x] = ax − xa. In
particular we ask when can the iterate of an inner derivation be an inner
derivation? When can the iterates of two derivations commute? More precisely, we
characterize elements a,b ∈ R, R a prime ring, for which (ada)n(x) = (adb)(x)
for all x ∈ R, and we characterize derivations d : R → R, δ : R → R for
which [dn(x),δn(y)] = 0 for all x,y ∈ R, R prime. Applications are made to
C∗-algebras.
