DIFFERENTIAL IDENTITIES WITH GENERALIZED SKEW DERIVATIONS ON PRIME NEAR RINGS

Abstract

In this thesis, we introduced the notion of generalized skew derivation in near-ring N and investigated several differential identities involving generalized skew derivations of a prime near-ring N which force N to be a commutative ring. First important preliminary concepts, definitions, examples, lemmas and theorems were presented to make the concept of generalized skew derivation in near rings clear. In addition, near rings definitions and properties of near rings are given which are helpful to study generalized skew derivation in prime near rings were discussed. This thesis is concerned with the generalized skew derivations on prime near rings and commuting conditions on generalized skew derivations in near rings. We proved that partially distributive laws of both skew derivation which is associated with automorphism 𝑔 on 𝑁 and generalized skew derivation associated with a non-zero skew derivation 𝐷 which is associated with automorphism 𝑔 on 𝑁. Also, we showed that differential identities with generalized skew derivation on near ring and generalized skew derivation in the setting of semigroup ideals of near rings to be commutative ring. And also we showed that primeness condition of commutativity of prime near rings admitting generalized skew derivation and generalized skew derivation in the setting of semigroup ideals were presented. Furthermore, we provide some examples to show that the restrictions imposed on the hypothesis of the various theorems are not superfluous. Finally, we showed that generalized skew derivations 𝐹 which acts as a homomorphism or an antihomomorphism on a semigroup ideal of a prime near rings shows that 𝐷 is zero or 𝐹 is identity map on 𝑁 and 𝑁 is a commutative ring