product of injective modules is injective

PropositionPlanetmathPlanetmath. Let R be a ring and {Qi}iI a family of injectivePlanetmathPlanetmath R-modules. Then the productPlanetmathPlanetmathPlanetmath


is injective.

Proof. Let B be an arbitrary R-module, AB a submoduleMathworldPlanetmath and f:AQ a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. It is enough to show that f can be extended to B. For iI denote by πi:QQi the projection. Since Qi is injective for any i, then the homomorphism πif:AQi can be extended to fi:BQi. Then we have


It is easy to check, that if aA, then f(a)=f(a), so f is an extensionPlanetmathPlanetmathPlanetmath of f. Thus Q is injective.

Remark. Unfortunetly direct sumMathworldPlanetmathPlanetmathPlanetmath of injective modulesMathworldPlanetmath need not be injective. Indeed, there is a theorem which states that direct sums of injective modules are injective if and only if ring R is NoetherianPlanetmathPlanetmath. Note that the proof presented above cannot be used for direct sums, because f(b) need not be an element of the direct sum, more precisely, it is possible that fi(b)0 for infinetly many iI. Nevertheless products are always injective.

Title product of injective modules is injective
Canonical name ProductOfInjectiveModulesIsInjective
Date of creation 2013-03-22 18:50:17
Last modified on 2013-03-22 18:50:17
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Theorem
Classification msc 16D50