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Title of Thesis

Mohammad Rashid Kamal Ansari
Institute/University/Department Details
Department of Mathematics/ University of Karachi
Number of Pages
Keywords (Extracted from title, table of contents and abstract of thesis)
cotorsion modules, ore domains, associative rings, left ore domains, torsion modules, torsion free modules, divisible modules, i-functors, d-rich functors

Matlis [16] has discussed cotorsion modules over integral domains whereas Henderson and orzech [12] has discussed cotorsion modules over commutative rings. In this thesis we emphasize non-commutative situations. We study cotorsion modules over associative rings and give particular attention to cotorsion modules over ore domains. With reference to an arbitrary ring R which is embedded in a ring D, we introduce D-cotorsion modules. Our results regarding the connection between D-cotorsion modules and modules appearing as Ext (A,B) generalize the results of matlis [16], Henderson and Orzech [12], Diana Yun-Dee Wei [30] and Qureshi [23]. Moreover, we generalize the work of matlis [16] regarding the links of torsion-cotorsion duality and cotorsion-cotorsion free duality with A-A* duality. Besides this we study the cotorsion completion functors of matlis [16] in the non-commutative case. Furthermore, we show that the category of cotorsion modules, which is a category with kernels and cokernels in case of integral domains, is a category with kernels and cokernels and cokernels in case of two sided oreβ domains also. In additions, a characterization of left ore domain in terms of cotorsion modules is provided.

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682.3 KB
S. No. Chapter Title of the Chapters Page Size (KB)
1 0 Contents
135.03 KB
2 1 Introduction 1
196.24 KB
  1.1 Preliminaries 11
  1.2 Left Ore Domains 11
  1.3 Dual Modules 12
  1.4 Torsion, Torsion Free And Divisible Modules 20
  1.5 Exact Sequences Of Matlis 34
3 2 D-Torsion And D- Cotorsion Modules 41
111.67 KB
  2.1 D-Torsion Modules 41
  2.2 D-Cotorsion Modules 45
4 3 Cotorsion Modules Over Ore Domains 58
203.86 KB
  3.1 Homomorphic Image Of A Torsion Free And Divisible 58
  3.2 Cotorsion Modules 67
  3.3 Cotorsion Free Modules 72
  3.4 Characterization Of Left Ore Domain 84
5 4 I-Functors And D-Rich Functors 89
179.93 KB
  4.1 Definitions And Examples 89
  4.2 Natural Transformation Of I-Functors 95
  4.3 References 112