The minimal ring which extends the semi-ring of natural numbers

Question

That's the definition of integers given by Encyclopedia of Math. What does it mean for a ring to be minimal? How to show its uniqueness?

Answer 1

Intuitively, it means that if $\mathbb N\subseteq R$ in the sense that $R$ contains (a copy of) the natural numbers as a subsemiring, then $\mathbb N\subseteq\mathbb Z\subseteq R$: that is, there's a copy of the integers in $R$.

In other words, $\mathbb Z$ is the smallest ring generated by $\mathbb N$. Obviously by using additive inverses alone, the smallest ring containing $\mathbb N$ would have to contain $-\mathbb N\cup \mathbb N$ (I include $0$ in the natural numbers). Moreover, you can verify that $-\mathbb N\cup \mathbb N$ is an additive abelian group that is closed under multiplication, so it is a ring (the ring of integers.)

To convince yourself of uniqueness, you should just realize that there is no choice in the matter of how to extend $\mathbb N$. The $1\in \mathbb N$ is in every other (unital) subring, and by using addition alone you generate $-\mathbb N\cup\mathbb N$. You can formalize it by using this observation.