A note on characterization of higher derivations and their product

‎There exists a one to one correspondence between higher derivations \{d_n\}_{n=۰}^\infty on an algebra \mathcal{A} and the family of sequences of derivations \{\delta_n\}_{n=۱}^\infty on \mathcal{A}‎. ‎In this paper‎, ‎we obtain a relation that calculates each derivation \delta_n (n \in \mathbb{N}) directly as a linear combination of products of terms of the corresponding higher derivation \{d_n\}_{n=۰}^\infty‎. ‎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 \mathcal{A}‎, ‎the product of them‎, ‎is a higher derivation on \mathcal{A}‎. ‎Also‎, ‎we prove that the product of two inner higher derivations‎, ‎is an inner higher derivation‎.

Keywords:

derivation‎ , ‎higher derivation‎ , ‎inner higher derivation