$S^{-1}R$ is local then it is of the form $R_P$ for some prime ideal $P$

Question

If $S^{-1}R$ is a local ring then prove that $S^{-1}R$ is of the form $R_P$ for some prime ideal $P$. (Here $S^{-1}R$ is denoting the localization of ring $R$ with respect to multiplicative closed set $S$.)

Answer 1

The only maximal ideal of $S^{-1}R$ is of the form $S^{-1}P$ with $P$ prime and $P\cap S=\emptyset$. Then $(S^{-1}R)_{S^{-1}P}\simeq S^{-1}R$. On the other side, $(S^{-1}R)_{S^{-1}P}\simeq R_P$.