Tests for injectivity of modules over commutative rings

TL;DR

This paper provides new criteria for determining when modules over commutative rings are injective, using Ext functors and flatness conditions, with results applicable to both noetherian and certain non-noetherian rings.

Contribution

It introduces novel injectivity criteria based on Ext vanishing and flatness, extending characterizations to non-noetherian rings.

Findings

Injectivity characterized by Ext^i vanishing over prime ideals

Injectivity criteria involving faithfully flat modules

Extension of results to certain non-noetherian rings

Abstract

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