| Let S,R be rings,SCR a semidualizing bimodule and X a class of some finitely presented right R-modules.In this thesis,firstly,the notions of C-X-injective modules and C-X-flat modules are introduced.Some properties of these two classes of modules are studied and some equivalent characterizations for a ring being X-coherent are given in terms of C-X-injective modules and C-X-flat modules.Secondly,the Foxby equivalences associated to C-X-injective and C-X-flat modules are established,and some properties of C-X-injective and C-X-flat modules are further studied over commutative rings.Finally,the right derived functors of-(?)-is defined via proper right C-X-injective resolutions and proper right C-X-flat resolutions of modules,and the left derived functors of Hom(-,-)is defined via proper right(left)C-X-injective resolutions of modules.Some characterizations of relative C-X-injective dimensions and relative C-X-flat dimensions of modules are given by these two derived functors. |
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