How do the authors justify that $\mathbb{Z}$, by construction, is clearly minimal?

Question

I am reading Section 9. The Rational Numbers from textbook Analysis I by Amann/Escher, where there is a theorem:

My thought:

1. smallest means that If there is another domain with unity, $\mathbf{Z}$, such that $\mathbb{N} \subseteq \mathbf{Z}$ and the ring operations on $\mathbf{Z}$ restrict to the usual operations on $\mathbb{N}$, then there is an injective homomorphism from $\mathbb{Z}$ to $\mathbf{Z}$.

2. This ring is unique up to isomorphism means that If there are two smallest domains with unity, $\mathbf{Z}, \mathbf{Z'}$, such that $\mathbb{N} \subseteq \mathbf{Z}, \mathbb{N} \subseteq \mathbf{Z'}$ and the ring operations on $\mathbf{Z}, \mathbf{Z'}$ restrict to the usual operations on $\mathbb{N}$, then $\mathbf{Z}$ and $\mathbf{Z'}$ are isomorphic.

My questions:

1. Is my thought correct?

2. I can not justify the statement Since $\mathbb{Z}$, by construction is clearly minimal in the proof. Please elaborate more on this point.

Thank you for your help!

Answer 1

If there was a smaller ring it would have to contain the additive inverse of $1$ ($-1$ by definition). Then it must contain $-n$ for all $n\in\mathbb{N}$, by closedness. In this way one shows that the ring contains all elements of $\mathbb{Z}$.