On The Generalizations Of Injective Modules And Flat Modules

Injective modules and flat modules play very important roles in homologicaltheory. In this thesis,we discuss the injective and flat properties of modules and usethem to characterize the regular rings and PS-rings. In the first chapter,the conceptof injective modules is extended and adjoint weak injective modules are defined. bymeans of this concept, we obtain a equivalent condition of wealky injective module.Furthermore adjoint weak global dimension is defined,and Schanuet's lemma oninjective modules is extended.The main results are proved as follows:Theorem 1.17 Right R-module N is weak-injective if and only if N hasadjoint weak injective resolution T: 0→N→T₀→T₁→…T_n→T_n+1→…,implies that for any left R-module A, AWExt_T¹(A, N)=0.Theorem 1.21 N-Schanuel Lemma:Suppose that there are two exact sequence:A→Q₀→Q₁→…→Q_n-1→NA→Q'₀→Q'₁→…Q'_n-1→N'where Q_i, Q'_i(i=0, 1,…n-1) are all adjoint weak injective modules,then Q₀⊕Q'⊕Q₂……(?))Q'_o⊕Q₁⊕Q'₂….In the second chapter, we introduce essential-R-modules and investigate theirproperties. The main results are following.Theorem 2.19 Assume that M satisfy the following condition:(1)for any left ideal I, M is essential-R/I-modules(2)for any left ideal I, if L∩K=0 is maximal for any L(?)R, then L⊕I(?)Rthen M is essential-R-module. In the third section,we generalize flat modules to SOC-flat modules and studyit properties.Finally, we invetigate SOC-flat rings and PS-rings with SOC-flat,andobtain the following theorem:Theorem 3.13 The following conditions on a ring R are equivalent:(1)R is SOC-flat ring;(2)Every right R-module is SOC-flat;(3)R/SOC(R)is left flat;(4)For every minimal right, ideal II(?)R/SOC(R)=0;(5)For every minimal right ideal I,I·SOC(R)=I;Theorem 3.14 The following conditions on a ring R are equivalent:(1)R is a PS-ring;(2)(SOC(R))²=SOC(R);(3)R/SOC(R)is left flat;(4)Every right R-module is SOC-flat.