9. Completion Fractions Modules of Filtered Modules over Non-necessarily Commutative Filtered Rings

The chapter 'Completion Fractions Modules of Filtered Modules over Non-necessarily Commutative Filtered Rings' delves into the intricate world of non-commutative filtered rings and modules. It begins by defining the necessary structures and conditions for topological groups and filtered groups, and then proceeds to construct the completion of filtered groups, rings, and modules. The core of the chapter focuses on the localization of completion modules and rings, utilizing saturated multiplicative subsets that satisfy left Ore conditions. Notably, the chapter proves several theorems demonstrating the isomorphism of completion rings and modules with their fraction rings and modules, and it explores the properties of S-saturated submodules in completion modules. The chapter concludes with a thorough analysis of the bijections and isomorphisms between submodules of completion modules, highlighting the Noetherian and Artinian properties of these modules. Throughout, the chapter offers a rigorous and detailed exploration of these advanced mathematical concepts, making it a valuable resource for researchers and academics in the field.