Canonical ring map

Question

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at $p$ (fraction field of $A/p$). Is the following proof correct?: notice $\mathbf{Z}/\chi^{-1}(p)\subset A/p\subset k(p)$, hence these rings have the same characteristic. Now if $\chi^{-1}(p)=(n)$ for some natural $n$ then the above implies $n=\mathrm{char} \ \mathbf{Z}/(n)=\mathrm{char} \ k(p)$.

Answer 1

The proof is correct. $\phantom{----}$