COMMUTATIVITY OF NEAR-RINGS WITH DERIVATIONS SATISFYING CERTAIN DIFFERENTIAL IDENTITIES

Abstract

The aim of this project was to elaborate the conditions for near ring with derivations to be a commutative ring. This is because near ring did not satisfies all the properties of ring. This project studied the conditions under which a near ring becomes a commutative ring. Some conditions imposed certain derivation on near rings satisfying certain differential identities were employed to prove certain commutativity theorems. Important notions, basic definitions, examples, remarks, basic lemmas and theorems related to our study which are required for the development of the main results of this project are presented. The concepts of derivations on near ring and its generalizations were discussed to make idea clear. However, the most important among them was near ring with derivations satisfying certain differential identities. The commutativity of addition and multiplication of near rings satisfying some conditions involving derivations was obtained. Moreover, examples proving the necessity of the primeness condition were given. This project gives a leisurely introduction to the theory of near-rings and its main results suitable for readers with a little knowledge of commutativity of near-ring.