The Generalized Matrix Module

It is widely acknowledged that if we define a kind of operation to the given ring ,we can get a module structure .This thesis mainly discuss the generalized matrix module based on the generalized matrix ring .First, we define the generalized matrix module and discuss the g.m.exact sequence to obtain the g.m.Three Lemma, the g.m.Five lemma ,especially the exact sequence and the diagram chasing ,which is the basic operation in the generalized matrix module. Next we discuss the g.m.idempotent, and further study the trace, reject, (finitely) generated, (finitely) cogenerated , and the product of the g.m.module. Second, we explore the discussion on the simple module, the semisimple module, the Socle and the Radical of the generalized matrix ring .Moreover on the base of the chain conditions, we know the relationships among the g.m.finitely generated module, the g.m.finitely cogenerated module and the g.m.semisimple module .Last, on the study of the g.m.projective modules, we know the relations among the g.m. projective modules, g.m. generators, g.m.projective covers and the covariant functor. In terms of the duality between the projective modules and the injective modules , we obtain the relations among the g.m.injective modules, g.m.cogenerators, g.m.injective envelopes and the g.m. contravariant functor .The two modules are the basis of the homological algebra and the module theory.