k-Derivations and k-Homomorphisms of Gamma Rings

The notion of a gamma ring is a generalization of the concept of a classical ring. This attempt characterizes certain gamma rings with various types of k-derivations and k-homomorphisms. We determine the commutativity of prime gamma rings of characteristic not equal to 2 and 3 with k-derivations, left (and right) k-derivations and generalized k-derivations. We prove that every Jordan k-derivation (also, Jordan generalized k-derivation) of a gamma ring is a k-derivation (generalized k-derivation) of the same, if we consider the gamma ring as a (2-torsion free) prime, completely prime, semiprime, and completely semiprime gamma ring, under some suitable conditions (as necessary), respectively. On the other hand, we prove that every Jordan k-homomorphism of a gamma ring onto a 2-torsion free prime (also, completely prime) gamma ring is either a k-homomorphism or an anti-k-homomorphism. The analogous result is also proved for Jordan k-isomorphism of a gamma ring onto a 2-torsion free prime/completely prime gamma ring. Finally, we investigate what does happen if a k-derivation acts as a k-endomorphism and also as an anti-k-endomorphism of certain gamma rings and look what we have here.