UP整环上的u-平坦模

u-flat Modules over UP-domains

设R是UP整环.R-模M是u-平坦模,是指对任意u-单同态f:A→B,使得1f:M_RA→M_RB是u-单同态.建立函子上的u-长正合列,证明R-模M是u-平坦模当且仅当对任何u-正合列0→A→B→C→0,序列0→M_RA→M_RB→M_RC→0是u-正合列,当且仅当对R的任何极大u-理想m,M_m是平坦R_m-模,当且仅当对R的任何理想I,自然同态M_RI→IM是u-同构.最后证明若{A_i|i∈Γ}是M的u-平坦子模的正向系,其中Γ是定向集,则lim→Ai是u-平坦模.

Let R be a UP domain. An R-module M is called u-flat if for each u-monomorphism f： A→B,1f： M_RA→MRB is a u-monomorphism. Using the  functor,we establish a long u-exact sequence. We prove that an R-module M is u-flat if and only if the sequence 0→M_RA→M_RB→M_RC→0 is a u-exact sequence for each u-exact sequence 0→A→B→C→0,if and only if Mmis a flat module for each maxiaml u-ideal m of R,if and only if the natural homomophism M_RI→IM is u-isomorphism of each ideal I of R. Moreover,it is shown that if { Ai｜ i∈Γ} is a direct system of u-flat submodules of M over a direct index set Γ,then lim→Aiis u-flat.