Characterizations of Prime and Semiprime Gamma Rings with Derivations

Gamma ring was first introduced by N. Nobusawa as a generalization of a classical ring. W. E. Barnes generalized the definition of a gamma ring due to Nobusawa. Presently the gamma ring due to Barnes is known as a gamma ring and gamma ring due to Nobusawa is known as gamma ring in the sense of Nobusawa and is denoted by -ring. It is clear that every ring is a gamma ring and every -ring is also a -ring. Actually, W.E.Barnes, J.Luh and S.Kyuno studied the structures of -rings and obtained various generalizations analogous to the corresponding parts in ring theory. Afterwards a number of algebraists have determined a lot of fundamental properties of -rings to classify and extend numerous significant results in classical ring theory to -ring theory. This work, entitled "Characterizations of Prime and Semiprime Gamma Rings with Derivations and Lie Ideals", aims to characterize prime and semiprime -rings with various types of left derivations, derivations, generalized derivations, higher derivations, derivations on Lie ideals and (U,M)-derivations.