A note on characterization of higher derivations and their product

Abstract

‎There exists a one to one correspondence between higher derivations $\{d_n\}_{n=0}^\infty$ on an algebra $\mathcal{A}$ and the family of sequences of derivations $\{\delta_n\}_{n=1}^\infty$ 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 $\{d_n\}_{n=0}^\infty$‎. ‎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