Discuss Homework 1 here!

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One Response to “Discuss Homework 1 here!”

  1. stevengubkin Says:

    Let \phi: A \rightarrow B be a ring homomorphism. Show that \phi^{-1} takes elements of Spec(B) to elements of Spec(A).

    Proof

    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.

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