Universal property of natural number semi-ring

Question

I asked a question similar to the one I am about to ask, and I think I got a satisfactory answer. However, this time I have some more specific question.

Let a semiring $(R,+,\times)$ be an algebraic structure such that $(R,+)$ is a monoid with identity $0$, and $(R,\times)$ is a monoid with identity $1$. Further suppose that the distributive law holds, and $0x=x0=0$ for all $x\in R$. We can then see that $\mathbf{N}$, the natural numbers (with $0$), form a semiring with respect to ordinary addition and multiplication.

My question is as follows.

Suppose now you construct $\mathbf{Z}$, the ring of integers, from $\mathbf{N}$ as follows:

1. Define an equivalence relation $\sim$ on $\mathbf{N}\times\mathbf{N}$ such that $(a,b)\sim(\alpha,\beta)$ if and only if $a+\beta=b+\alpha$. Define addition $\oplus$ on $\mathbf{N}\times\mathbf{N}/\mathord{\sim}$ as \begin{equation}[(a,b)]\oplus[(\alpha,\beta)] = [(a+\alpha,b+\beta)],\end{equation} and multiplication $\otimes$ as \begin{equation}[(a,b)]\otimes[(\alpha,\beta)] = [(a\alpha+b\beta,a\beta+b\alpha)].\end{equation}

These are indeed well-defined functions on the quotient, and by identifying each element $[(a,b)]\in\mathbf{Z}$ (for $b>a$) as $b-a$ in $\mathbf{N}$, we have $\mathbf{Z}$. Call this map $i:\mathbf{N}\to\mathbf{Z}$. Now for the $real$ question:

Is it true that given any other ring $R$ that contains a homomorphic copy of $\mathbf{N}$, or equivalently, given any injective semiring homomorphism $\phi:\mathbf{N}\to R$, is it true that there exists a unique injective semiring homomorphism $\tilde{\phi}:\mathbf{Z}\to R$ such that $\phi=\tilde{\phi}\circ i$?

Much appreciated in advance!

Answer 1

It does not even seem to have anything to do with one-to-one morphisms.

$\Bbb N$ is clearly initial in the category of semirings: for a given semiring $S$, there is a unique semiring homomorphism from $\Bbb N \to S$ determined by $\phi(1)=1_S$. (The kernel could even be nonzero.)

Likewise, $\Bbb Z$ is initial in the category of rings, where the unique map from $\Bbb Z\to R$ is given by $\psi(1)=1_R$.

These two maps are fully determined by additivity and preservation of multiplicative identity, and the only difference is their domain. For each ring $R$, $\psi$ is necessarily the only ring homomorphism extending the semiring homomorphism $\phi$.