Show that the localization at p, prime ideal, $R_p$ is a field iff for any $x \in p$ there exists $y \not\in p$ such that $xy = 0$
I know there is a similar question where R is an Noetherian ring, but this question does not have any conditions like that. I know there still is a one to one correspondence with the prime ideal in $R_p$ and the prime ideal in $R$ contained in $p$. So the only valid prime ideal is $pR_p$? And also the nilrad(R_p) is the intersection of all prime ideal, so nilrad(R_p) = $pR_p$. But I don't know how to put this all together and show the both if and only if direction.