Semi-projective Modules And Kernel-generators

The notions of projective modules and generators are fundamental in ring theory,module theory and homological algebra.Projective modules have been generalized in various points of view,including quasi-projective and semiprojective modules.There are also some generaliza.tions of generators such as self-generators and kernel-generators.Modules M,which generate the kernel of every endomorphism of M,have been involved several times in the literature.For convenience,we refer to such modules as kernel-generators.Comparatively speaking,the known results on semiprojective modules and kernel-generators are limited.Inspired by previous related results,we carry out further resea.rch on semiprojective modules and kernel-generators.For semiprojective modules,we note that different terminologies were adopted in the literature,including self-projective,image-projective,quasi-principally projec-tive.On the other hand,the terminology "semiprojective,has different meaning in some different articles.We follow the definition that a module M is called semipro-jective if,for any endomorphism α and γ with γ(M)(?)α(M),there exists an endo-morphism α such that γ=αβ.For the sake of distinction,"Doman-projective" and"weakly quasi-projective" are used for the other two kinds of generalized projectivity We start with some examples to distinguish some notions related to semiprojective modules.For instance,semiprojective modules versus(weakly)quasi-projective and Doman-projective modules.Then three equivalent characterizations of semiprojec-tive modules are given from different aspects in terms of precovers of modules,homo-topic chain maps between complexes and defining matrices of modules,respectively As applications,we characterize semisimple and(semi)perfect rings by semiprojec-tive modules.M-principally projective modules are also investigated since they are closely related to semiprojective modulesThe second part of this thesis is devoted to kernel-generators.We introduce the notion of(m,n)-kernel-generators,where m are n fixed but arbitrary positive integers.Note that all self-generators and generalized morphic modules are kernel-generators.We have examples which demonstrate that a kernel-generator need not be a self-generator or a generalized morphic module.It is also proved that the direct summand of a kernel-generator need not be a kernel-generator.Thus submodules and homomorphic images of kernel-generators need not be kernel-generators.For a generalized morphic module(in particular,a quasi-morphic module)M,we obtain further equivalent condition under which M is semiprojective.In addition,pseudo-morphic properties of a module M and its endomorphism ring are studied under the condition that M is semiprojective and is a kernel-generator,respectively.