COTORSION MODULES OVER ORE DOMAINS

Abstract

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.