A note on characterization of higher derivations and their product

There exists a one to one correspondence between higher derivations {dn}n∞=0 on an algebra A and the family of sequences of derivations {δn}n∞=1 on A. In this paper, we obtain a relation that cal-culates each derivation δn(n ∈ N) directly as a linear combination of products of terms of the corresponding higher derivation {dn}n∞=0. Also, we find the general form of the family of inner derivations corresponding to an inner higher derivation. We show that for every two higher derivations on an algebra A, the product of them is a higher derivation on A. Also, we prove that the product of two inner higher derivations is an inner higher derivation.