The Study Of Some Properties Of W-FP_n- Flat And W-FP_n- Injective Modules

In this thesis,we mainly study the w-module theoretic analogue of FP_n-flat modules and FP_n-injective modules in terms of w-module theory over commutative rings,where n is a non-negative integer or ∞.Chapter 2 consists of a summary of some basic concepts and result from finiteness conditions in modules and w-module theory.In Chapter 3,for a non-negative integer n,we introduce and investigate w-FP_n-flat and w-FP_n-injective modules over commutative rings.Firstly,some basic properties of them are discussed.As an application,we give a new characterization of semi-simple rings,that is,a commutative ring R is semi-simple if and only if every R-module is w-FP₀-injective.Moreover,we discuss the relationship between w-FP_n-flat and w-FP_n-injective modules by introducing generalized character modules.It is shown that an R-module M is w-FP_n-flat(w-FP_n-injective)module if and only if its generalized character module M+ is w-FP_n-injective(w-FP_n-flat)module.In Chapter 4,we introduce the notions of w-weak flat and w-weak injective modules over commutative rings,namely w-FP_∞-flat and w-FP_∞-injective modules.Firstly,we discuss several general properties and characterizations of these modules.Secondly,we introduce the w-weak flat dimension and w-weak injective dimension of modules.Several characterizations of them are given.Finally,we also introduce and study the w-super finitely presented dimension of a commutative ring that is defined in terms of only super finitely presented modules.We show that the w-super finitely presented dimension of a commutative ring R is 0 if and only if every R-module is w-weak flat,if and only if every R-module is w-weak injective,if and only if every super finitely presented R-module is w-split.