On (alpha, beta)-n-derivations and certain n-additive mappings in rings with involution

Abstract

Let R be an associative ring with involution ∗. In this paper we introduce the notion of (α,β)∗-n-derivation in R, where α and β are endomorphisms of R. An additive mapping x↦x∗ of R into itself is called an involution on R if it satisfies the conditions: (i) (x∗)∗=x, (ii) (xy)∗=y∗x∗ for all x,y∈ R. A ring R equipped with an involution ∗ is called a ∗-ring. In the present paper it is shown that if a ∗-prime ring R admits a nonzero (α,β)∗-n-derivation D such that α is surjective, then R is commutative. Some properties of certain n-additive mappings are also discussed in the setting of ∗-prime rings and semiprime ∗-rings. Further, some related properties of (α,β)∗-n-derivation in semiprime ∗-ring have also been investigated. Besides, we have also constructed several examples throughout the text to justify that various restrictions imposed in the hypotheses of our theorems are not superfluous. Finally a structure theorem for (α,β)∗-n-derivation in a semiprime ∗-ring has been established.