Weak Injective Modules And Weak Morita Duality

Injective modules play an important role in module theory and homological algebra. In the first part of this paper, the concept of injective modules extended and weak injective modules are denned. A module T is called weak injective in case for any R-module N, any non-zero submodule M of N, and any 0 ≠ a ∈ HomR(M, N) such that for any β ∈ HomR(M, T) with Kera Kerβ, there exists γ ∈ HomR(M,T) satisfying β = γa. We call a an adjoint homomorphism of M. On the base of the concept, we give the Schanual Lemma about weak injective modules and discuss the properties of the homomorphisms which can be extended in weak injective modules. Otherwise, we define the weak injective dimension and the weak global dimension and study the module whose weak injective dimension is 0 or 1.In [5], (m,n)-injective modules were defined. We call an R-module T a (m,n)-injective module in case every R-homomorphism from an n-generated submodule M of Rm to T extends to one from Rm to T. The (m,n)-injective modules unify the concept of FP-injective modules, f-injective modules and p-injective modules. In this paper, we give the notion of weak (m,n)-injective modules and study weak (m,n)-injective modules using of annihilator. Moreover, we define weak principally quasi-injective modules and study Jacobson radical of endomorphism ring of weak principally quasi-injective modules.Morita duality is an important concept in module theory. In [11],[12],[13],[14],[15], Morita duality is discussed.Mr.Zhu extended Morita duality to weak Morita duality in [12]. In the second part of this paper, using of linear compact and injective cogenerator so on, we discuss the relations between Morita duality and weak Morita duality.In [16], n-absolutely pure modules, n-flat modules and n-coherent rings are defined. Moreover, it was proved that if R is an n-coherent ring, then the direct product of n-flat R-modules is n-flat R-module. In this paper, we study the relations between n-absolutely pure modules and n-flat modules and define finite present n-flat modules and finite copresent n-absolutely pure modules.