example of injective module

In the categoryMathworldPlanetmath of unitary -modules (which is the category of Abelian groups), every divisible Group is injectivePlanetmathPlanetmath, i.e. every Group G such that for any gG and n, there is a hG such that nh=g. For example, and / are divisible, and therefore injective.


We have to show that, if G is a divisible Group, φ:UG is any homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and U is a subgroupMathworldPlanetmathPlanetmath of a Group H, there is a homomorphism ψ:HG such that the restrictionPlanetmathPlanetmathPlanetmath ψ|U=φ. In other words, we want to extend φ to a homomorphism HG.

Let 𝒟 be the set of pairs (K,ψ) such that K is a subgroup of G containing U and ψ:KG is a homomorphism with ψ|U=φ. Then 𝒟 ist non-empty since it contains (U,φ), and it is partially ordered by

(K,ψ)(K,ψ):KK and ψ|K=ψ.

For any ascending chain


in 𝒟, the pair (iKi,iψi) is in 𝒟, and it is an upper bound for this chain. Therefore, by Zorn’s Lemma, 𝒟 contains a maximal element (M,χ).

It remains to show that M=H. Suppose the opposite, and let hHM. Let h denote the subgroup of H generated by h. If hM={0}, the sum M+h is in fact a direct sumMathworldPlanetmathPlanetmath, and we can extend χ to M+h by choosing an arbitrary image of h in G and extending linearly. This contradicts the maximality of (M,χ).

Let us therefore suppose hM contains an element nh, with n minimalPlanetmathPlanetmath. Since nhM, and χ is defined on M, χ(nh) exists, and furthermore, since G is divisible, there is a gG such that ng=χ(nh). It is now easy to see that we can extend χ to M+h by defining χ(h):=g, in contradictionMathworldPlanetmathPlanetmath to the maximality of (M,χ).

Therefore, M=H. This proves the statement. ∎

Title example of injective module
Canonical name ExampleOfInjectiveModule
Date of creation 2013-03-22 17:43:40
Last modified on 2013-03-22 17:43:40
Owner Glotzfrosch (19314)
Last modified by Glotzfrosch (19314)
Numerical id 5
Author Glotzfrosch (19314)
Entry type Example
Classification msc 16D50