A ring R is called left pure semisimple if very left R-module is a direct sum of finitely generated modules. This thesis based on [15] investigates preinjective modules over pure semisimple rings. For a left pure semisimple ring R, preinjective left R-modules provide very useful information on the category of finitely generated left R-modules. A finitely presented indecomposable left module M is called preinjective if the family R-ind has a cofinite subset H, such that there is ni monomorphsm M?N, N add H. the concept of For a left pure semisimple ring R, it is shown that the local duality establishes a bijection between the preinjective left R-modules and the preprojective right R-modules. We showe for a left pure semisimple ring R, any direct sum of preinjective modules is endoartinian.Any preinjective left R-modules is the source of a left almost split morphism. We prove for pure semisimple ring R, and subcategory of R-Mod such that is closed under direct sums and submodules, contains only finitely many non-isomorphic indecomposable almost splitting injective modules. We prove our main results on the subcategory of left R-modules cotaning no preinjective direct summands, and give applications for determining the indecomposable direct summands of products of preinjective left R-modules. Let R be any left pure semisimple ring, and the subcategory of all left R-modules with ni preinjective direct summands. Then contains only finitely many non-preinjective indecomposable modules almost splitting injective modules. If there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R-mod, the direct sum of all non-preinjective indecomposable direct summand of products of preinjective left R-modules is a finitely generated product-complete module.