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

Proposition^{}. Abelian group^{} $A$ is divisible if and only if $A$ is an injective object in the category of abelian groups.

Proof. ,,$\Leftarrow $” Assume that $A$ is not divisible, i.e. there exists $a\in A$ and $n\in \mathbb{N}$ such that the equation $nx=a$ has no solution in $A$. Let $$ be a cyclic subgroup generated by $a$ and $i:B\to A$ the canonical inclusion. Now there are two possibilities: either $B$ is finite or infinite^{}.

If $B$ is infinite, then let $H=\mathbb{Z}$ and let $f:B\to H$ be defined on generator^{} by $f(a)=n$. Now $A$ is injective^{}, thus there exists $h:H\to A$ such that $h\circ f=i$. Thus

$$n\cdot h(1)=h(1)+\mathrm{\cdots}+h(1)=h(1+\mathrm{\cdots}+1)=h(n)=h(f(a))=i(a)=a.$$ |

Contradiction^{} with definition of $n\in \mathbb{N}$ and $a\in A$.

If $B$ is finite, then let $k=|B|$ (note that $n$ does not divide $k$) and let $H={\mathbb{Z}}_{n\cdot k}$. Furtheremore define $f:B\to H$ on generator by $f(a)=n$ (note that in this case $f$ is a well defined homomorphism^{}). Again injectivity of $A$ implies existence of $h:H\to A$ such that $h\circ f=i$. Similarly we get contradiction:

$$n\cdot h(1)=h(1)+\mathrm{\cdots}+h(1)=h(1+\mathrm{\cdots}+1)=h(n)=h(f(a))=i(a)=a.$$ |

This completes^{} first implication^{}.

,,$\Rightarrow $” This implication is proven here (http://planetmath.org/ExampleOfInjectiveModule). $\mathrm{\square}$

Remark. It is clear that in the category of abelian groups $\mathcal{A}\mathcal{B}$, a group $A$ is projective if and only if $A$ is free. This is since $\mathcal{A}\mathcal{B}$ is equivalent^{} to the category^{} of $\mathbb{Z}$-modules and projective modules^{} are direct summands^{} of free modules^{}. Since $\mathbb{Z}$ is a principal ideal domain^{}, then every submodule of a free module is free, thus projective $\mathbb{Z}$-modules are free.

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