## Discuss Homework 1 here!

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Let $\phi: A \rightarrow B$ be a ring homomorphism. Show that $\phi^{-1}$ takes elements of $Spec(B)$ to elements of $Spec(A)$.
Let $\mathfrak{b} \in Spec(B)$ and let $\pi: B/\mathfrak{b}$ be the canonical projection. Then it is easy to see that $\phi^{-1}[\mathfrak{b}]$ is the kernel of $\pi \circ \phi$. Thus $A/\phi^{-1}[\mathfrak{b}] \cong Im(\pi \circ \phi)$. But $Im(\pi \circ \phi)$ is a subring of an integral domain (because $\mathfrak{b}$ is prime), so it is an integral domain. Thus $\phi^{-1}[\mathfrak{b}]$ is a prime ideal in $A$.