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.

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Sujoy Chakraborty was born in 1972 in Comilla, Bangladesh. He earned the M.Sc. degree in Pure Mathematics in 1993 and the M.Phil. degree in 2002 at the University of Dhaka. He was awarded the Ph.D. degree in 2010 for this dissertation at the University of Rajshahi. He has been teaching at Shahjalal University of Science and Technology since 1999.