整环上的u-算子及其同调特征

u-Operation over an Integral Domain and Its Homological Characterizations

通过U-内射模定义了UP整环以及UP整环上的u-算子和u-模,证明了UP整环上,M是U-挠模当且仅当对任何正合列0→A→B→M→0,其中B是U-内射模,有A_u=B;也证明了M是U-内射模当且仅当同态f可以扩张到A_u,当且仅当对任何U-挠模C,Ext_R^1(C,M)=0.其次,在UP整环上定义了u-正合列,证明了A→fB→gC是u-正合列当且仅当(im(f)+ker(g))/im(f)与(im(f)+ker(g))/ker(g)都是U-挠模.最后,在UP整环上证明了若A→fB→gC→0是u-正合列,N是u-模,则0→Hom_R(C,N)→Hom_R(B,N)→Hom_R(A,N)是正合列.

We define an UP domain,u-operation and u-modules by using U-injective modules.We prove that,over an UP-do-main,M is U-torsion if and only if for each exact sequence 0 →A →Bg→ M→O,where B is a U-injective module,A u =B holds.We also prove that M is a U-injective module if and only if a morphism f：A →M can be extended to A u,if and only if for each U-torsion module C,Ext 1R （C,M） =0.Moreover,we give the definition of u-exact sequences over a UP-domain and prove that A^f →B^g →C is a u-exact sequence if and only if both（im（f） ＋ ker（g））/im（f） and （im（f） ＋ ker（g））/ker（g） are U-torsion modules.Finally,we show that over a UP-domain,if A ^f→B^g→C →O is a u-exact sequence and N is a u-module,then O →HomR （C,N） →Hom R （B,N） →HomR （A,N） is an exact sequence.