Research On Flat Objects In-left R-module Category Of ?-set

Category theory as a whole to study the relations of space form and quantity of contact and transformation,revealing the external characteristics of things well.In the study of modern mathematics,category theory played an important role in many fields.In this paper,we introduce the concept of flat object in the category of ?-left R-modules,e?uivalent descriptions of an flat object in the category of ?-left R-modules is investigated,a product of a set of ?-left R-modules is an flat object if and only if every one is an flat object is proved,a free ?-left R-module,project ?-left R-module and inject ?-left R-module was proved to be a flat ?-left R-module.In this paper,based on the study of the related properties of flat die,further enrich the research contents of flat die,to make it more perfect,hope you can after the mathematical foundation of category theory and on category and a collection of research have certain reference value.