SEMIGROUP IDEALS AND COMMUTATIVITY OF NEAR RINGS WITH SEMIDERIVATIONS SATISFYING CERTAIN DIFFERENTIAL IDENTITIES

Abstract

The main purpose of this project is to elaborate the ideas and concepts of semigroup ideals and commutativity of near rings with semiderivations satisfying certain differential identities. The important preliminary concepts, definitions, lemmas and theorems were elaborated with detailed explanatory steps to make the concept clear. In addition, to study commutativity of near rings definitions and properties of semiderivation and semigroup ideals were employed. Detailed explanatory steps on the proofs of the theorems and lemmas on commutativity of near rings admitting a semiderivation satisfying certain differential identities were provided. Also, we have seen the cases that in some results the primeness, -torsion free and zero symmetric of near ring in the setting of semigroup ideals involving semiderivations are sufficient condition for commutativity of near ring. Further in the main results of this study detailed proofs by including more explanatory steps to show commutativity of near ring under the assumptions “a -torsion free prime near ring that admits a nonzero semiderivation satisfying ( ) is contained in the multiplicative center of ( )” and “a prime near ring with semiderivation acting as a homomorphism or anti-homomorphism in the setting of a semigroup ideal of .”