Quotient of Rationals by Integers is Injective

Theorem

Let $\struct {\Q, +}$ be the abelian group of rational numbers.

Let $\struct {\Z, +}$ be the abelian group of integers, considered as a subgroup of $\struct {\Q, +}$.

Then the quotient group $\Q / \Z$ is an injective object in the category of abelian groups.

Proof

By definition, $\struct {\Q, +, \times}$ is the field of quotients of the ring of integers $\struct {\Z, +, \times}$.

By Field of Quotients is Divisible Module‎ $\Q$ is a divisible $\Z$-module.

By Quotient of Divisible Module is Divisible $\Q / \Z$ is a divisible $\Z$-module.

By Ring of Integers is Principal Ideal Domain $\struct {\Z, +, \times}$ is a principal ideal domain.

By Injective Module over Principal Ideal Domain $\Q / \Z$ is an injective $\Z$-module.

Since the category of abelian groups is equivalent to the category of $\Z$-modules, $\Q / \Z$ is an injective object in the category of abelian groups.

$\blacksquare$