Derivatives on various algebraic structures and effects of derivative on algebraic structures

This thesis deals with generalized versions of the concept of derivative on algebraic structures, especially on prime rings and prime near rings, and systematically analyzes the effects of these transformations on the related structures. First, the basic concepts that will provide an understanding of the subject are examined. Then, important works in the literature on Jordan derivative, symmetric Jordan bi-derivative, symmetric left Jordan bi-derivative transformations defined on prime rings are given. One of the original contributions of the research is to show that the concept of derivative has been carried beyond the classical ring theory and defined in the context of near rings, which are algebraic structures with fewer axioms. In this context, the properties of the derivative defined on near rings and prime near rings are analyzed in detail with theorems and proofs. In this context, the existence conditions of differential transformations defined on near rings and their relations with central structures are mathematically revealed. By emphasizing the differences of near rings from classical rings, the flexibility offered by these structures in algebraic analysis is evaluated at the theoretical level. In the main part of the thesis, a skew symmetric Jordan bi-derivative transformation on prime rings is defined. The criteria provided by the defined derivative transformation on the ring are analyzed in detail. In the last part of the thesis, a symmetric Jordan bi- (α, β) derivative on prime rings is defined, and similar criteria is proved for this derivative transformation. In conclusion, this thesis provides a comprehensive and theoretically sound framework for the characterization of algebraic structures by studying the structural effects of differential transformations on prime and near-prime rings in a multifaceted way. Keywords: Prime ring, near ring, Jordan derivative, skew symmetric Jordan bi-derivative, symmetric Jordan bi- (α, β) derivative