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.
On Generalized Jordan Triple (α, β)*-Derivations and Related Mappings
Let R be a 2-torsion free semiprime *-ring and let α, β be surjective endomorphisms of R. The aim of the paper is to show that every generalized Jordan triple (α, β)*-derivation on R is a generalized Jordan (α, β)*-derivation. This result makes it possible to prove that every generalized Jordan triple (α, β)*-derivation on a semisimple H*- algebra is a generalized Jordan (α, β)*-derivation. Finally, we prove that every Jordan triple left α*-centralizer on a 2-torsion free semiprime ring is a Jordan left α*-centralizer.
A Note on Derivations II
In a previous note on derivations [1] we determined the structure of a prime ring R which has a derivation d≠0 such that the values of d commute, that is, for which d(x) d(y) = d(y) d(x) for all x, y∈R. Perhaps even more natural might be the question: what elements in a prime ring commute with all the values of a non-zero derivation? We address ourselves to this question here, and settle it.