abelian group is divisible if and only if it is an injective object

PropositionPlanetmathPlanetmathPlanetmath. Abelian groupMathworldPlanetmath A is divisible if and only if A is an injective object in the category of abelian groups.

Proof. ,,” Assume that A is not divisible, i.e. there exists aA and n such that the equation nx=a has no solution in A. Let B=<a> be a cyclic subgroup generated by a and i:BA the canonical inclusion. Now there are two possibilities: either B is finite or infiniteMathworldPlanetmath.

If B is infinite, then let H= and let f:BH be defined on generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath by f(a)=n. Now A is injectivePlanetmathPlanetmath, thus there exists h:HA such that hf=i. Thus


ContradictionMathworldPlanetmathPlanetmath with definition of n and aA.

If B is finite, then let k=|B| (note that n does not divide k) and let H=nk. Furtheremore define f:BH on generator by f(a)=n (note that in this case f is a well defined homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath). Again injectivity of A implies existence of h:HA such that hf=i. Similarly we get contradiction:


This completesPlanetmathPlanetmathPlanetmathPlanetmath first implicationMathworldPlanetmath.

,,” This implication is proven here (http://planetmath.org/ExampleOfInjectiveModule).

Remark. It is clear that in the category of abelian groups 𝒜, a group A is projective if and only if A is free. This is since 𝒜 is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the categoryMathworldPlanetmath of -modules and projective modulesMathworldPlanetmath are direct summandsMathworldPlanetmath of free modulesMathworldPlanetmathPlanetmath. Since is a principal ideal domainMathworldPlanetmath, then every submodule of a free module is free, thus projective -modules are free.

Title abelian group is divisible if and only if it is an injective object
Canonical name AbelianGroupIsDivisibleIfAndOnlyIfItIsAnInjectiveObject
Date of creation 2013-03-22 18:48:15
Last modified on 2013-03-22 18:48:15
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 20K99