**Exercise 5.2.23**

We demonstrate categories with maps that are epic but *not* surjective:

**Theorem**:
In the category of monoids , the inclusion is epic

*Proof*:
Suppose we have maps and .
This essentially means that and are identical on positive integers.
We show that this implies that they’re also identical on negative integers.
Indeed, let .
Then,

then, since , we have :

as desired. Since and are clearly identical on , we have for all , so .

**Theorem**:
In the category of rings , the inclusion is epic.

*Proof*:
Suppose we have maps for some ring , and .
This means that and are identical on the integers.
We show this implies they are identical on all rationals.
Indeed, let be some fully reduced rational number.
Then, we have:

and , so

and so,

as desired. So for all rational , and .