On the Structure of Modules Defined by Opposites of FP Injectivity

Abstract

Let R be a ring with unity and let $$M_R$$MR and $$_RN$$RN be right and left modules,respectively. The module $$M_R$$MR is said to be absolutely $$_RN$$RN-pure if $$M \\otimes N \\rightarrow L \\otimes N$$M⊗N→L⊗N is amonomorphism for every extension $$L_R$$LR of $$M_R$$MR. For a module $$M_R$$MR, the subpurity domain of $$M_R$$MR is defined to be the collection of all modules $$_RN$$RN, such that $$M_R$$MR is absolutely $$_RN$$RN-pure. Clearly, $$M_R$$MR is absolutely $$_RF$$RF-pure for every flat module $$_RF$$RF and that $$M_R$$MR is FP-injective if the subpurity domain of M is the entire class of left modules. As an opposite of FP-injective modules, $$M_R$$MR is said to be a 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. We characterize the structure of t.f.b.s. modules over commutative hereditary Noetherian rings. We prove that a module M is t.f.b.s. over a commutative hereditary Noetherian ring if and only if M\xa0/\xa0Z(M) is t.f.b.s. if and only if $${\\text {Hom}}(M/Z(M), S) \\ne 0$$Hom(M/Z(M),S)≠0 for each singular simple module S. Prüfer domains are characterized as those domains all of whose nonzero finitely generated ideals are t.f.b.s.