Rings and modules characterized by opposites of FP-injectivity
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Let $R$ be a ring with unity. Given modules $M_R$ and $_RN$, $M_R$ is said to be absolutely $_RN$-pure if $M \otimes N \to L \otimes N$ is a monomorphism for every extension $L_R$ of $M_R$. For a module $M_R$, the subpurity domain of $M_R$ is defined to be the collection of all modules $_RN$ such that $M_R$ is absolutely $_RN$-pure. Clearly $M_R$ is absolutely $_RF$-pure for every flat module $_RF$, and that $M_R$ is FP-injective if the subpurity domain of $M$ is the entire class of left modules. As an opposite of FP-injective modules, $M_R$ 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. $R_R$ 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.
Abstract
Let $R$ be a ring with unity. Given modules $M_R$ and $_RN$, $M_R$ is said to be absolutely $_RN$-pure if $M \otimes N \to L \otimes N$ is a monomorphism for every extension $L_R$ of $M_R$. For a module $M_R$, the subpurity domain of $M_R$ is defined to be the collection of all modules $_RN$ such that $M_R$ is absolutely $_RN$-pure. Clearly $M_R$ is absolutely $_RF$-pure for every flat module $_RF$, and that $M_R$ is FP-injective if the subpurity domain of $M$ is the entire class of left modules. As an opposite of FP-injective modules, $M_R$ 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. $R_R$ 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.
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