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Full-Text Articles in Physical Sciences and Mathematics
Generalized Derivations Centralizing On Jordan Ideals Of Rings With Involution, Lahcen Oukhtite, Abdellah Mamouni
Generalized Derivations Centralizing On Jordan Ideals Of Rings With Involution, Lahcen Oukhtite, Abdellah Mamouni
Turkish Journal of Mathematics
A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. In this paper we extend Posner's result to generalized derivations centralizing on Jordan ideals of rings with involution and discuss the related results. Moreover, we provide examples to show that the assumed restriction cannot be relaxed.
Generalized Derivations On Jordan Ideals In Prime Rings, Mahmoud El-Soufi, Ahmed Aboubakr
Generalized Derivations On Jordan Ideals In Prime Rings, Mahmoud El-Soufi, Ahmed Aboubakr
Turkish Journal of Mathematics
Let R be a 2-torsion free prime ring with center Z(R), J be a nonzero Jordan ideal also a subring of R, and F be a generalized derivation with associated derivation d. In the present paper, we shall show that J\subseteq Z(R) if any one of the following properties holds: (i) [F(u), u]\in Z(R), (ii) F(u)u = ud(u), (iii) d(u^2)=2F(u)u, (iv) F(u^2)-2uF(u) = d(u^2)-2ud(u), (v) F^2(u)+3d^2(u)=2Fd(u)+2dF(u), (vi) F(u^2) = 2uF(u) for all u \in J.