On derivations and commutativity in prime near-rings

Identities With Additive Mappings in Semiprime Rings

“…Furthermore, Brešar [7] proved that every additive commuting mapping of a prime ring R is of the form x → λx + ζ(x), where λ is an element of the extended centroid C and ζ : R → C is an additive mapping. For results concerning commuting mappings, centralizing mappings and related problems we refer the reader to [1,[5][6][7][8][9][10][11][12][13]18,[22][23][24][25][26][27][28] where further references can be found.…”

On Commutativity of Semiprime Rings with Multiplicative (Generalized)-derivations

“…In the literature, a number of authors have discussed the commutativity of prime rings and semiprime rings admitting derivations and generalized derivations satisfying certain algebraic identities, see (Ali, Kumar & Miyan, 2011), (Ali, Dhara & Fosner, 2011), (Andima & Pajoohesh, 2010), (Ashraf et al, 2007(Ashraf et al, , 2001, (Daif & Bell, 1992), (Dhara & Pattanayak, 2011), (Hongan, 1997), where further references can be found.…”

On multiplicative (generalized)-derivations in semiprime rings

“…As a …rst time, in Ashraf and Rehman's paper [7], if d(xy) xy 2 Z(R) holds for all x; y 2 I, then R is commutative where R is a prime ring, I is nonzero two sided ideal of R and d : R ! R is a derivation.…”