On ∗-commuting mappings and derivations in rings with involution

Abstract

Let R be a ring with involution ∗. A mapping f : R → R is said to be ∗-commuting on R if [f(x), x∗ ] = 0 holds for all x ∈ R. The purpose of this paper is to describe the structure of a pair of additive mappings that are ∗- commuting on a semiprime ring with involution. Furthermore, we study the commutativity of prime rings with involution satisfying any one of the following conditions: (i) [d(x), d(x ∗ )] = 0, (ii) d(x) ◦ d(x ∗ ) = 0, (iii) d([x, x∗ ]) ± [x, x∗ ] = 0 (iv) d(x ◦ x ∗ ) ± (x ◦ x ∗ ) = 0, (v) d([x, x∗ ]) ± (x ◦ x ∗ ) = 0, (vi) d(x ◦ x ∗ ) ± [x, x∗ ] = 0, where d is a nonzero derivation of R. Finally, an example is given to demonstrate that the condition of the second kind of involution is not superfluous.