Tests for injectivity of modules over commutative rings

Abstract

It is proved that a module $ M$ over a commutative noetherian ring $ R$ is injective if $ \mathrm{Ext}_{R}^{i}((R/{\mathfrak p})_{\mathfrak p},M)=0$ for every $ i\ge 1$ and every prime ideal $ \mathfrak{p}$ in~$ R$ . This leads to the following characterization of injective modules: If $ F$ is faithfully flat, then a module $ M$ such that $ \Hom_R(F,M)$ is injective and $ \Ext^i_R(F,M)=0$ for all $ i\ge 1$ is injective. A limited version of this characterization is also proved for certain non-noetherian rings.