G_V-injective Modules And Projectively Coresolved Gorenstein AC-flat Modules Over Triangular Matrix Rings

This paper mainly studies Gv-injective modules,special GV-injective preenvelopes and projectively coresolved Gorenstein AGflat modules(i.e.PGACF modules)over triangular matrix ring T=(?).It is proved that(1)Let(V1,V2)∈ AMod × B-Mod,M=(?)∈ T-Mod.If V=h(V1,V2)is Π-self-orthogonal,U is V-cocompatible,and X2 ∈ ProdB(V2),(?)∈⊥1ProdT(V),then M is GV-injective left T-module if and only if kerφM is GV1-injective left A-module,M2 is GV2-injective left B-module,and φM:M1→ HomB(U,M2)is an epimorphism.(2)Let(V1,V2)∈ A-Mod × B-Mod,X2∈ProdB(V2),(?)∈⊥1ProdT(V).If V=h(V1,V2)is Π-self-orthogonal,V1 is weakly cotilting,and so is V2,U is Vcocompatible,and GVI(T)is special preenveloping in T-Mod,then GV1I(A)is special preenveloping in A-Mod,and Gv2I(B)is special preenveloping in B-Mod.(3)Let rAC-d(B)<∞.If UA is finitely generated projective module,BU is flat module,and M=(?)is PGACF left T-module,then M1 is PGACF left A-module,M2/ImφM is PGACF left B-module,and φM:U(?)B M1→M2 is a monomorphism;If the class of PGACF modules is close under extensions,then the opposite case holds.