The Generalizations Of Homological Modules And Its Properties

This paper consists of four chapters. Chapter 1 is introduction,we introduce the importance of homological theory in Algebras and the close relationships with other algebraic branchs.In the second chapter,we extend the conception of right f-injective from ring to module and obtain the notion of right fm-injective module. In this chapter,we give some characterizations of fm-injective modules. We also derive various equivalent conditions for a ring to be fm-coherent.In the third chapter,we introduce the concept of generalized projective module.A right R-module M is called Generalized projective,if for every epimorphism g : N→L in the right R-module category and every homomorphism f : M→L, there exists h : M→N such that Im(f-gh)<< L.It is proved that a generalized projective module when it has a projective cover is a projective;And we character ize hereditary rings in terms of generalized projective modules, the endomorphism ring of generalized projective modules is also discussed.In the fourth chapter,we introduce the concept of maximal flat module and develope its some properties,and derive the relations between maximal flat module,flat module and maximal injective module.Moreover,we obtain the relation between max-flat dimension and global dimension.Our main results as follows:Theorem 2.2.4 Let R be a ring and _RM is a fm-injective module , _RN≤_RM,π: M→M/N natural epimorphism,then the following are equivalent:(1) _RN is fm-injective module;(2) For every finite generated submodule I of R^m,g∈Hom_R(R^m/I, M/N), there exists h∈Hom_R(R^m/I, M) such that g=πh;(3) g(I) = N_m + r_Mm,write N(I) = {x∈M_m|Ix (?) N},for every finite generated submodule I of R^m.Theorem 3.2.5 If M is a generalized projective module, when it has a projective coverφ: P→M,then M is a projective module.Theorem 3.2.9 The following statements are equivalent:(1) R is a right hereditary ring; (2) All submodules of generalized projective modules are projective.Theorem 4.2.19 Let R be a ring,R is left perfect ring if and only if every left maximal fiat module is projective.