On the Structure of Some Types of Higher Derivations

Abstract

In this paper we introduce the concepts of higher {Lgn , Rhn }-derivation, higher {gn, hn}-derivation and Jordan higher {gn, hn}-derivation. Then we give a characterization of higher {Lgn , Rhn }-derivations and higher {gn, hn}-derivations in terms of {Lg, Rh}-derivations and {g, h}-derivations, respectively. Using this result, we prove that every Jordan higher {gn, hn}-derivation on a semiprime algebra is a higher {gn, hn}-derivation. In addition, we show that every Jordan higher {gn, hn}- derivation of the tensor product of a semiprime algebra and a commutative algebra is a higher {gn, hn}-derivation. Moreover, we show that there is a one to one correspondence between the set of all higher {Lgn , Rhn }-derivations and the set of all sequences of {LGn , RHn }-derivations. Also, it is presented that if A is a unital algebra and {fn} is a generalized higher derivation associated with a sequence {dn} of linear mappings, then {dn} is a higher derivation. Some other related results are also discussed.