GENERALIZED SKEW HOMODERIVATIONS IN PRIME RINGS

Originating from the study of noncommutative polynomials vanishing under all substitutions from a given algebra, the classical theory of PIs has evolved into a rich framework that connects combinatorial and structural aspects of algebraic systems, with matrix algebras playing a paradigmatic role. A natural extension of this theory is provided by generalized polynomial identities (GPIs), in which coefficients may lie within the algebra itself. Since Amitsur's foundational work and Martindale's generalization to prime rings, GPIs have been investigated in increasingly broad contexts, incorporating involutions, automorphisms, derivations, and anti-automorphisms, thereby yielding deep characterizations of prime and semiprime structures. Within this lineage, derivations and their many generalizations, such as generalized derivations, skew derivations and generalized skew derivations, have been a persistent source of commutativity criteria and structural constraints. Since the second half of the twentieth century, numerous authors have investigated generalized skew derivations that satisfy specific generalized polynomial identities on non-central Lie ideals of prime or semiprime rings. Particularly notable is the concept of \emph{homoderivation}, introduced by El-Sofy, which unifies aspects of endomorphisms and derivations. Recent developments, including $\epsilon$-homoderivations, have expanded this idea, but the full structural potential of such mappings remains largely unexplored. This dissertation advances the theory by introducing the notion of \emph{extended homoderivations}, additive maps from a prime ring $R$ into its right Martindale ring of quotients, associated with scalar parameters in the extended centroid. These maps, together with their generalized, skew, and extended skew variants, subsume and unify classical concepts such as derivations, generalized derivations, homomorphisms, and their skew analogues. The study systematically develops the theory of these mappings, extending and reformulating cornerstone results of Posner, Herstein, and others, and establishing new commutativity theorems in this broadened setting. Special attention is devoted to the interaction between these mappings and structural components such as Lie ideals, centralizing conditions, and torsion-free hypotheses. The dissertation is organized to progressively build this framework: beginning with generalized homoderivations, it extends to the fully skew and extended contexts, culminating in a classification of the most general class considered, the \emph{extended generalized skew homoderivations}. Applications illustrate the breadth of the theory, including analyses of subrings generated by images of the form $F(x)x$ and structural constraints arising from generalized Jordan-type identities.\\ More specifically, in the course of our analysis, we have undertaken a broad classification of the maps introduced, thereby providing a comprehensive and multifaceted picture of this emerging framework. Furthermore, we have presented an application that integrates various concepts and elucidates the precise structure of two generalized skew derivations, $F$ and $G$, in the context where they act as a natural generalization of Jordan homoderivations.\\ \noindent In summary, this work situates extended homoderivations and their variants as a unifying theme in the theory of additive mappings on prime and semiprime rings, demonstrating how classical results on derivations naturally extend to a richer, more flexible setting. By bridging and generalizing multiple strands of existing theory, it provides new tools and perspectives for the structural analysis of noncommutative rings.