The Direct Limits Of Modules About The Properties Of Localization

Direct limits play an important role in the algebra, and the method of localization is classical in researching the properties of modules, so these two directions are studied extensively. About the localized construction and existence of the direct limits, and the exchangeability for the operator S-1and lim,â†'there have been some good conclusions in the current literature. But, about the relation between the localized direct systems and its corresponding direct limits, generalized localization and the exact properties of the direct limits, we study less. So in this paper, we meanly study the properties of the localized direct systems and its corresponding direct limits.Firstly, this paper studies what the direct limits of the localized direct systems S-1lim Mi about the fractional modules S-1Mi (i∈I) of direct systems are maximal flat modules, single modules,loyal moudles. Then we introduce adjoint systems of the commutative rings and commutative modules, and prove that localized commutative rings and commutative modules is also adjoint. Further, we prove the diagram exchangeability of the localized direct limits.Secondly, for the direct limits of the modular direct systems, we obtain its generalized localization form on the basic knowledge about its classical localization, and prove that generalized localization of the modles of direct limits of the modular direct systems are also modules, generalized localization of the direct systems are also direct systems. Then, we prove the universal property and embedded property of the generalized localization of the direct limits.Finally, we introduce the tensor product representation of the the direct limits of the localized modular direct systems, then prove the exact properties of the localized direct limits.