Ring Monomorphism from Integers to Rationals

Theorem
Let $\phi: \Z \to \Q$ be the mapping from the integers $\Z$ to the rational numbers $\Q$ defined as:
 * $\forall x \in \Z: \phi \left({x}\right) = \dfrac x 1$

Then $\phi$ is a (ring) monomorphism, but specifically not an epimorphism.

Proof
First note that:
 * $\forall a, b \in \Z: a \ne b \implies \dfrac a 1 = \phi \left({a}\right) \ne \phi \left({b}\right) = \dfrac b 1$

and so clearly $\phi$ is an injection.

However, take for example $\dfrac 1 2$:
 * $\not \exists a \in \Z: \phi \left({a}\right) = \dfrac 1 2$

as $\dfrac 1 2 \notin \operatorname {Im} \left({\phi}\right)$.

So $\phi$ is not a surjection.

Next let $a, b \in \Z$.

So $\phi$ can be seen to be an injective, but not surjective, ring homomorphism.

Hence the result by definition of ring monomorphism and ring epimorphism.