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 Jordan Triple α-*Centralizers Of Semiprime Rings
Abstract Let R be a 2-torsion free semiprime ring equipped with an involution *. An additive mapping T : R→R is called a left (resp. right) Jordan α-*centralizer associated with a function α: R→R if T(x2)=T(x)α(x*) (resp. T(x2)=α(x*)T(x)) holds for all x ∊ R. If T is both left and right Jordan α-* centralizer of R, then it is called Jordan α-* centralizer of R. In the present paper it is shown that if α is an automorphism of R, and T : R→ R is an additive mapping such that 2T(xyx)=T(x) α(y*x*) +α(x*y*)T(x) holds for all x, y ∊ R, then T is a Jordan α-*centralizer of R
On Semicommuting Automorphisms of Rings
Abstract Let R be a prime ring possessing a nontrivial automorphism T such that xTx = ±xxT. If R is not of characteristic 3 or R has nonzero center, then R is a commutative integral domain.