Let $R$ be a commutative ring with unity and let $\mathfrak p$ be a prime ideal of $R$. Let $S= R \setminus \mathfrak p$. Then $R_{\mathfrak p} = S^{-1} R$ is a local ring. To what extent is the converse true? I mean the following:
If $S^{-1} R$ is a local ring for some multiplicative subset $S \subset R$, then is $S^{-1} R = R_{\mathfrak p}$, for some prime ideal $\mathfrak p$?
Maybe the same answer as the previous one, but shorter. If $S^{-1}R$ is local, then its maximal ideal looks like $S^{-1}P$ with $P$ prime and $P\cap S=\emptyset$. Then $S^{-1}R\simeq (S^{-1}R)_{S^{-1}P}\simeq R_P$.