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!
Best Answer
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.