Gorenstein X-projective Modules And Dimension

Since Auslander and Bridge gave the concept of Gorenstein dimension of finitely generated module,many researchers began their research on modules of Gorenstein di-mension zero,especially the study of Gorenstein projective modules by Enochs,Holm,etc makes the branch of algebra rapid development.Inspired by the researcher’s achieve-ments above.In this paper,for a class X of modules that contains all projective modules,Gorenstein X-projective modules and Gorenstein X-projective dimensions are inves-tigated,and some of their properties are given.In particular,we prove that Gorenstein X-projective syzygy modules are exactly projective syzygy modules.The first chapter is an introduction to introduce the background of the paper and some definitions,theorems and lemmas,etc.In the second chapter,we give the definition of Gorenstein X-projective modules and discuss some properties of Gorenstein X-projective modules,and illustrate the re-lationship with Gorenstein projective modules,Ding projective modules and Gorenstein AC-projective modules.In the third chapter,we discuss the stability of Gorenstein X-projective modules,and prove that the nth Gorenstein X-projective modules are Gorenstein X-projective modules.In the fourth chapter,we discuss Gorenstein X-projective dimensions,where we prove that Gorenstein X-projective syzygy modules are exactly projective syzygy mod-ules.In the fifth chapter,using the methods of Bennis,Mahdou and other researchers on strongly Gorenstein projective modules,we discuss some properties of strongly Goren-stein X-projective modules.