TRACES OF PERMUTING 3–GENERALIZED DERIVATIONS AND COMMUTATIVITY IN PRIME NEAR RINGS

Abstract

The main purpose of this study is to elaborate the conditions for a near ring with permuting 3- generalized derivation to be a commutative ring. Permuting 3-generalized derivations in near rings can tell us about the structure of the near rings. This project studied the conditions under which a near ring becomes commutative ring. Some conditions imposed on traces of permuting 3-additive maps, traces of permuting 3-derivations and 3-generalized derivations were employed to prove certain commutativity theorems. Important preliminary notions, basic definitions, examples and some important well known results related to the development of the main results of this project are presented. In order to facilitate discussions we presented how it is possible to extend some results for near rings with traces of permuting 3-derivations to near rings with traces of permuting 3-generalized derivations. The concepts of different types of derivations and their generalizations on near rings were discussed to make the ideas clear. However, the most important among them was the notions of permuting 3-generalized derivations in near rings. The commutativity of addition and multiplication of near rings satisfying some conditions involving permuting 3-right generalized derivations was obtained. Finally, we presented about what happens if we take semigroup ideal instead of near ring in the commutativity of prime near ring. Moreover, we provided more detail explanatory steps in the proof of some commutativity theorems in the setting of a semigroup ideal of a prime near ring admitting permuting 3-generalized derivation and also give illustrations to justify the notions in various theorems.