I am going through these lecture notes, and I am confused about the proof of Corollary 10.10 there.

Corollary 10.10. Let $A$ be a local noetherian domain with maximal ideal $\mathfrak{p}$, let $g \in A[x]$, and let $B := A[x]/g(x)$. Every maximal ideal $\mathfrak{m}$ of $B$ contains the prime $\mathfrak{p}B$.

In the proof of this, the author writes

The ring $B$ is finitely generated over the noetherian ring $A$, hence a noetherian $A$-module, ...

I don't understand this part, because I think if $g$ is not monic, then $B$ will not be finitely generated? For example, take $A = \mathbb{Z}_{(3)}$, and $g(x) = 3 \in A[x]$. Then $B := A[x]/g(x) = \mathbb{Z}_{(3)}[x]/3\mathbb{Z}_{(3)}[x] = (\mathbb{Z}/3\mathbb{Z})[x]$ which is certainly not finitely generated over $A$? I'm not sure what I am missing here. If there is another source for the proof of this corollary, please let me know. Thank you so much.

  • $\begingroup$ You cannot use g=3=0 for the quotient. If g is not zero, because you are in a filed, you can assume that g is monic. $\endgroup$
    – Bongo
    Jun 10 at 8:39
  • 1
    $\begingroup$ @Bongo Local ring Z_(3) does have unique maxinal ideal 3Z_(3), though. (Edit) I also do not get what this material is trying to convey.. $\endgroup$
    – Abastro
    Jun 10 at 8:42
  • $\begingroup$ My hunch is that deg g > 0 is implied. $\endgroup$
    – Abastro
    Jun 10 at 9:02
  • 2
    $\begingroup$ This happens when one learns mathematics from lecture notes instead of books. $\endgroup$
    – user26857
    Jun 10 at 10:10
  • $\begingroup$ @user26857 I tried to learn from Cassels and Frohlich but found it to be too dry for self-studying. Would you recommend me some books that cover similar material as these notes? $\endgroup$ Jun 10 at 17:08


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