Resumen de Tests for injectivity of modules over commutative rings

Lars Winther Christensen, Srikanth B. Iyengar

• 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 holds for every i\geqslant 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 {\text {Hom}}_R(F,M) is injective and {\text {Ext}}^i_R(F,M)=0 for all i\geqslant 1 is injective. A limited version of this characterization is also proved for certain non-noetherian rings.