On the Structure of Some Types of Higher Derivations

Amin Hosseini ; Nadeem Ur Rehman

Kragujevac Journal of Mathematics, Tome 48 (2024) no. 1, p. 123

Résumé

In this paper we introduce the concepts of higher $\{L_{g_n}, R_{h_n}\}$-derivation, higher $\{g_n, h_n\}$-derivation and Jordan higher $\{g_n, h_n\}$-derivation. Then we give a characterization of higher $\{L_{g_n}, R_{h_n}\}$-derivations and higher $\{g_n, h_n\}$-derivations in terms of $\{L_g, R_h\}$-derivations and $\{g, h\}$-derivations, respectively. Using this result, we prove that every Jordan higher $\{g_n, h_n\}$-derivation on a semiprime algebra is a higher $\{g_n, h_n\}$-derivation. In addition, we show that every Jordan higher $\{g_n, h_n\}$-derivation of the tensor product of a semiprime algebra and a commutative algebra is a higher $\{g_n, h_n\}$-derivation. Moreover, we show that there is a one to one correspondence between the set of all higher $\{L_{g_n}, R_{h_n}\}$-derivations and the set of all sequences of $\{L_{G_n}, R_{H_n}\}$-derivations. Also, it is presented that if $\mathcal{A}$ is a unital algebra and $\{f_n\}$ is a generalized higher derivation associated with a sequence $\{d_n\}$ of linear mappings, then $\{d_n\}$ is a higher derivation. Some other related results are also discussed.

Détail

Classification : 16W25, 47B47
Keywords: higher $\{L_{g_n}, R_{h_n}\}$-derivation, higher $\{g_n, h_n\}$-derivation, Jordan higher $\{g_n, h_n\}$-derivation, generalized higher derivation