Some Researches Of Generalized Projective Modules

Projective modules are the important modules in modules theory and coherent algebra theory, and also have a lot of applications in algebraic geometry study. so some generalizations on this module is studied and discussed in this article, it is divided into three chapters.In the first chapter,it is extend the concept of projective module and introduce the concept of weak projective module and discuss the Schanuel lemma on weakly projec-tive module,then define the weak projective dimension and weak global dimension,and the characteristics of the modules are discussed for weak projective dimension0and1, secondly, based on the weak projective module, define the adjoint weak projective module,and also get some properties of the adjoint weak projective module.In the second chapter, on the basis of the concept of M-principally projective module,Firstly,define pseudo-M-principally projective module, give some equivalent conditions of pseudo-M-principally projective module,and discuss its some basic prop-erties.Then give the concept of quasi pseudo principally projective module,further dis-cussing the endomorphism ring of quasi pseudo principally projective module,and proof that under certain conditions the endomorphism ring of quasi pseudo principally projec-tive module is semiprimary ring.Lastly,using quasi pseudo principally projective mod-ule to portray semisimple ring and perfect ring.The third chapter firstly promote the concept of the small pseudo projective mod-ule, introducing small M-pseudo projective module,Then discuss some properties of small M-pseudo projective module,proof that under certain conditions module N is small M-pseudo projective module equivalent to N is M-pseudo projective module.