SOME RESULTS ON GENERALIZED REVERSE DERIVATIONS ON PRIME NEAR RINGS WITH SEMIGROUP IDEALS

Abstract

The principal purpose of this thesis is to extend results from reverse derivation prime rings to generalized reverse derivations on prime near rings in the setting of semigroup ideals and discuss conditions under which a prime near ring is commutative ring. The extensions and conditions are not easily generalized, because one of a classical problem in near ring theory is to study and generalize the conditions under which conditions a prime near ring is commutative ring. First important preliminary concepts, examples, lemmas and theorems are presented. In order to study generalized reverse derivations on prime near rings with semigroup ideals with some terms such as derivations, reverse derivations, generalized derivations are discussed. The concept of generalized reverse derivations in prime near rings with semigroup ideals and some important results of extension of generalized reverse derivations in prime near rings with semigroup ideals were presented and we have showed that certain conditions involving generalized reverse derivations forces a prime near ring to be a commutative ring. Moreover, we have proved that a non-zero generalized reverse derivation (f, d) on N with semigroup ideals U of N such that f acts as a homomorphism and an anti-homomorphism on semigroup ideal U of N, thus f is the identity map on semigroup ideals U of N. Finally, we have proved that product of two generalized reverse derivations act as generalized reverse derivation on semigroup ideal ideals of N, such that at least one of generalized reverse derivation is zero. For further study researchers can do on multiplicative (generalized) reverse derivations on prime near ring.