Example of a ring $R$ and a prime $0\neq P\unlhd R$ such that $R_P$ is a field?

Question

Give an example of a commutative ring $R$ and a nonzero prime ideal $P\unlhd R$ such that $R_P$ is a field.

I can't seem to think of any examples. If anyone has ideas, I would also appreciate hints on how you found your example (maybe briefly outline your thought process?), and also your method of showing that the localization at that prime is indeed a field.

Answer 1

We know that $R_{\mathfrak p}$ is a local ring with $\mathfrak p^e$ being the maximal ideal, so for $R_{\mathfrak p}$ to be a field, we must have $\mathfrak p^e = 0$, i.e. $\frac x 1 = 0$ for every $x \in \mathfrak p$, i.e. for every $x \in \mathfrak p$ there is $s \notin \mathfrak p$ such that $xs = 0$.

An example that comes to mind is $2 \times 3 = 0$ in $\Bbb Z/6\Bbb Z$, so $R = \Bbb Z/6\Bbb Z$ and $\mathfrak p = \langle 2 + 6 \Bbb Z \rangle$, in which case $R_{\mathfrak p} \cong \Bbb F_2$, the field of $2$ elements.

Answer 2

Such a prime ideal $\mathfrak p$ is necessarily a minimal prime ideal, , hence an associated prime ideal, i.e. if the ring $R$ is noetherian, $\mathfrak p=\operatorname{Ann}_R a$. If $a\notin \mathfrak p$, it is a unit in $R_{\mathfrak p}$, so $\mathfrak pR_{\mathfrak p}=\{0\}$, which means $R_{\mathfrak p}$ is a field.

Example :

Take $R=K[X,Y]/(XY)$, $\mathfrak p=(X)/(XY)$ is the annihilator of $Y+(XY)$, and $$k(\mathfrak p)\simeq K(Y).$$