On(X,Y,Z)-Gorenstein Homological Dimensions

In this paper,we use a triad(X,Y,Z)and a pair(X,Y)of modules to study relative Gorenstein homology theory respectively,obtain(X,Y,Z)-Gorenstein projective and(X,Y,Z)-Gorenstein injective modules and dimensions,and weak(X,Y)Gorenstein modules.In the first chapter,we mainly investigate the class GP(X,Y,Z)of(X,Y,Z)Gorenstein projective modules and the class GI(X,Y,Z)of(X,Y,Z)-Gorenstein injective modules,obtain their some equivalent characterizations,and give some closeness of GP(X,Y,Z)and GI(X,Y,Z)under(finite)direct sums,direct products,direct summands,extensions,kernels of epimorphisms and cokernels of monomorphisms.In particular,GP(X,P,Z)is resolving and GI(L,Y,Z)is coresolving.We also study the stability of GP(X,Y,Z)and GI(X,Y,Z),and obtain some conditions of the equations GPn(X,Y,Z)=GPn+1(X,Y,Z)and GIn(X,Y,Z)=GIn+1(X,Y,Z).In particular,for the known classes GP(P,P,P)and GI(L,L,L)of Gorenstein projective modules and Gorenstein injective modules,we have the equations GP2(P,P,P)=GP(P,P,P),GI2(L,L,L)=GI(L,L,L).In the second chapter,we introduce and investigate(X,Y,Z)-Gorenstein homology dimensions of modules and rings.We define(X,Y,Z)-Gorenstein projective dimensions of modules by a(X,Y,Z)-Gorenstein projective resolution,obtain some equivalent characterizations of(X,Y,Z)-Gorenstein projective dimensions by several kinds of resolutions of modules,give some relations between left Y-resolutin dimensions and(X,Y,Z)?Gorenstein projective dimensions,and use Ext-functors to characterize modules with finite(X,Y,Z)-Gorenstein projective dimensions.We also introduce(X,Y,Z)-Gorenstein projective right global dimensions of rings,and study some properties of a ring R with GP(X,Y,Z)—pd(R)?1.Dually,we introduce the(X,Y,Z)-Gorenstein injective dimension of modules and rings,and give duality conclusions correspondingly.We prove that(?GI(X,Y,Z),GI(X,Y,Z)is a cotorsion pair,and then every right R-module has a special GI(X,Y,Z)-preenvelope.We also prove that GI(X,Y,Z)-id(R)=0 if and only if each right R-module has GI(X,Y,Z)-preenvelope with unique mapping property if and only if each right Rmodule has a monic GI(X,Y,Z)-cover.Under an almost excellent extension of rings,we obtain some invariant properties of(X,Y,Z)-Gorenstein projective and injective modules and give some equations of(X,Y,Z)-Gorenstein projective and injective dimensions of modules and rings.In the third chapter,we introduce and study weak(X,Y)-Gorenstein modules,obtain some conditions that every weak(X,Y)-Gorenstein module is(X,Y,Z)Gorenstein projective(injective)modules.We investigate the stability of the class wG(X,Y)of weak(X,Y)-Gorenstein modules,and obtain some conditions of the equations Y=wGn(X,Y)and X=wGn(X,Y).In particular,P=wG(P,P)and L=wG(L,L)whenever rgD(R)<?,while wG(L,P)=P if and only if R is a QF ring if and only if wG(L,P)=L.Under a Morita duality,we show that weak(X,Y)-Gorenstein modules and weak(Y*,X*)-Gorenstein modules constitute a dual pair.As special kinds of weak(X,Y)-Gorenstein modules,we study some properties of weak(X,Y)-Gorenstein projective and injective modules.Finally,we obtain several equivalent characterizations that every right R-module is weak(X,Y)-Gorenstein,weak(X,Y)-Gorenstein projective and injective respectively.As a corollary,we obtain some known equivalent characterizations of QF rings.