Small R-projective Modules And ?-quasi-projective Modules

The thesis is divided into five chapters.In chapter 1,we introduce the history of modules and the important role in the development process on the algebra it played.In chapter 2,we introduce some related concepts and related lemmas.In chapter 3,the concepts of small N-projective modules and small R-projective modules are given.A right R-module M is called a small N-projective module if for every epimorphism f:N?N/N₁ with N₁ is a small submodule of N and every homomorphism g:M?N/N₁,there exists a homomorphism h:M?N such that fh=g.A right R-module M is called a small R-projective module,if M is small RR-projective Basic properties of these modules are studied.It is proved that a ring R is semiprimitive if and only if every right R-module is small R-projective.Meanwhile,the concept of small R-projective covers is posed.An epimorphism f:P?M?or P?is called a small R-projective cover of M,if P is small R-projective and f is a superfluous epimorphism.It is proved that a ring R is semiperfect if and only if R/J?R?is semisimple and every simple R-module has a small R-projective cover;a ring R is right perfect if and only if R/J?R?is semisimple and every semisimple R-module has a small R-projective coverIn chapter 4,we introduce the concepts of ?-R-projective modules and essentially R-projective modules.Let ? be a set of right ideals of R.M is called a?-R-projective module if for every canonical epimorphism ?:R?R/X with X??and every homomorphism g:M?R/X,there exsits a homomorphism h:M?R such that?h=g.M is called an essential R-projective module if M is ?-R-projective where?={I/I is an essential right ideal class of R}.we study basic properties of r-R-projective modules,It is proved that a ring R is semisimple if and only if every right R-module is essentially R-projective.In chapter 5,the concepts of ?-quasi-projective modules and small quasi-projective modules are given.Let ? be a set of the submodules of M.M is called a ?—quasi-projective module if for every canonical epimorphism?:M?M/X with X?? and every homomorphism g:M?M/X,there exsits a homomorphism h:M?M such that ?h=g.M is called a small quasi-projective module if M is ?-quasi-projective where ?={N|N is a small submodule class of M}.Some properties of ?-quasi-projective modules and small-quasi-projective modules are investigated.