Rings and modules characterized by opposites of FP-injectivity

摘要

Let R be a ring with unity. Given modules MR and RN, MR is said to be absolutely RN-pure if M?N→L?N is a monomorphism for every extension LR of MR. For a module MR, the subpurity domain of MR is defined to be the collection of all modules RN such that MR is absolutely RN-pure. Clearly MR is absolutely RF-pure for every flat module RF, and that MR is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, MR is said to be a emph{test for flatness by subpurity (or t.f.b.s.~for short)} if its subpurity domain is as small as possible, namely, consisting of exactly the flat left modules. Every ring has a right t.f.b.s.~module. RR is t.f.b.s.~and every finitely generated right ideal is finitely presented if and only if R is right semihereditary. A domain R is Pr"{u}fer if and only if R is t.f.b.s. The rings whose simple right modules are t.f.b.s.~or injective are completely characterized. Some necessary conditions for the rings whose right modules are t.f.b.s.~or injective are obtained.