Some Special Gorenstein Modules And Their Applications

Gorenstein modules are the very important objects in relative homological alge-bra.The g(x,y,l)-modules[40]generalize g(x)-modules[35],strongly Goren-stein flat modules[15],Gorenstein FP-injective modules[23,30]and so on.In this the-sis,we mainly study strongly(x,y,l)-Gorenstein modules(Sg(x,y,l)-modules for short)and their applications,g(x,y,l)-resolution dimensions and the homo-logical properties of n-Sg(x,y,l)-modules,where n is a positive integer.This thesis is divided into four chapters.In Chapter 1,main results and preliminaries are stated.In Chapter 2,we introduce strongly(x,y,l)-Gorenstein modules,where x,y,l are additive full subcategories of R-Mod.These modules provide a new characterization of g(x,y,l)-modules,i.e.,a module is a g(x,y,l)-module if and only if it is a direct summand of a certain strongly(x,y,l)-Gorenstein mod-ule under some additional conditions.Another application which concerns the global dimensions is given:the left global g(x,y,l)-resolution dimension is equal to the right global g(x;,y',l')-resolution dimension over any ring R,where x',y',l'are additive full subcategories of R-Mod related to x,y,l.In Chapter 3,we prove that x is a generator and a cogenerator for g(x,y,l).We give a characterization of the g(x,y,l)-resolution dimension of M ? R-Mod,i.e.,assume y?x and x?l,if the x-resolution dimension of M is finite,then it equals the g(x,y,l)-resolution dimension of M.In Chapter 4,we introduce n-Sg(x,y,l)-modules,where x,y,l are addi-tive full subcategories of R-Mod and n is a positive integer.They generalize Sg(x,y,l)-modules,n-strongly Gorenstein projective and injective modules[10,43]and so on.Assume x(?)y and x(?)l,then we give some characterization-s of n-Sg(x,y,l)-modules and some relations between n-S9(x,y,l)and m-Sg(x,y,l).