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Let $A$ be a commutative ring with identity and $\mathfrak{q}\subset\mathfrak{p}$ two prime ideals of $A$. I am trying to determine whether $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ is an integral domain.

Original attempt: Since $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}\cong(A/\mathfrak{q})_\mathfrak{p}$ and the latter is the localization of an integral domain, $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ should be integral too.

Trouble: I noticed afterwards that the isomorphism above is an isomorphism of $A_\mathfrak{p}$-modules but not rings. Also, $(A/\mathfrak{q})_\mathfrak{p}$ is not the localization of $A/\mathfrak{q}$ as a ring but as a $A$-module, because $\mathfrak{p}$ is not an ideal of $A/\mathfrak{q}$ but of $A$. So I cannot conclude that $(A/\mathfrak{q})_\mathfrak{p}$ is an integral domain (or even a ring).

Question:

(1) Is $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ an integral domain?

(2) Does $(A/\mathfrak{q})_\mathfrak{p}$ have a natural and well-defined ring structure? I am guessing it does because of the isomorphism, and I assume that the ring structure of $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ passed on to it is the structure I want. But how can I justify this?

Any help is greatly appreciated. Thanks in advance!

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    $\begingroup$ You say that "$p$ is not an ideal of $A/q$ but of $A$". But $p/q$ is an ideal of $A/q$. $\endgroup$ – Angina Seng May 25 '19 at 3:57
  • $\begingroup$ @LordSharktheUnknown But I am not doing localization of $A/\mathfrak{q}$ at $\mathfrak{p}/\mathfrak{q}$. $\endgroup$ – trisct May 25 '19 at 4:04
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    $\begingroup$ And for showing that $A_p/q_p$ is an integral domain it's enough to notice that $q_p$ is a prime ideal. (The form of prime ideals of $A_p$ is well known.) $\endgroup$ – user26857 May 25 '19 at 7:22
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You have the isomorphism $A_p=A\otimes_A A_p$. Now you get that $A_p/qA_p = A/q\otimes_A A_p=(A/q)_p$ and this gives you the ring structure, because the tensor product has a canonical structure of algebra over $A$. Then as you say it is easy to see that $A_p/qA_p$ is a domain, since $A/q$ is a domain and a localization of a domain is a domain.

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