On Jordan and Jordan*-Generalized Derivations in Semiprime Rings with Involution
The purpose of this note is to prove the following result. Let R be a6 −torsion free semiprime *-ring and let G: R−→R be an additive mapping such that G(xyx )= G(x)y ∗ x ∗ + xD(y)x ∗ + xyD(x) holds for all x,y ∈ R and some *-derivations D of R. Then G is a Jordan*generalized derivation.
JORDAN TRIPLE(α,β)*-DERIVATIONS ONSEMIPRIME RINGS WITH INVOLUTION
Let R be a 2-torsion free semiprime ∗-ring. The aim of this paper isto show that every Jordan triple (α, β)∗ -derivation on R is a Jordan(α, β)∗ -derivation. Furthermore, every Jordan triple left α∗ -centralizeron R is a Jordan left α∗ -centralizer. Consequently, every generalizedJordan triple (α, β)∗ -derivation on R is a Jordan (α, β)∗ -derivation.
Jordan *-derivation pairs and quadratic functionals on modules over *-rings
We give a new simple proof of Semrl’s recent representation theorem for quasi-quadratic functions acting on unital modules and then show that our approach also gives a certain extension of Semrl’s result.