Determining structure of rings by means of their finitely annihilated modules

This thesis is based on work on modules that satisfy the H-condition, also known as "finitely annihilated modules" in the theory of modules on unitary rings. Modules that satisfy the H-condition have taken an important place in ring theory and attracted attention by many mathematicians because of their emergence and effective use in topics such as Homological Algebra and localization in non-commutative rings. The H-condition, believed to have been proposed by P. Gabriel [8] in the literature, allows a transition between the structure of the ring and the structure of the module on it. The purpose of this thesis is to reveal the structure, examples and importance of finitely annihilated modules, to examine the ring structure consisting of finitely annihilated modules on some module classes. The first chapter of this thesis, which consists of five chapters, consists of information about the historical development and importance of the thesis topic. The second chapter includes the basic definitions and theorems required in the next chapters. In the third chapter, finite annihilated modules are defined and the basic properties they provide are examined. In the fourth chapter, Artinian Rings are characterized by being finite annihilated of each module on it, and the concept of "weak H-condition" is defined. In the last chapter, the effects of semisimple modules, uniform modules, and injective modules to satisfy the H-condition on the ring structure are examined. Keywords: Ring, Module, Finitely Annihilated Module, H-condition, Artinian Ring, Semisimple Module, Uniform Module, Injective Module, Singular module