Let $\displaystyle P$be a prime ideal,$\displaystyle R$be a commutative ring with identity,$\displaystyle S=R-P$. Then $\displaystyle S$will be a multiplicative subset of $\displaystyle R$, that is, $\displaystyle 1\in S,a,b\in S\Rightarrow ab\in S$

Define $\displaystyle S^{-1}R$ as the set $\displaystyle R\times S$.

define $\displaystyle (a,s)=(a_1,s_1)\Leftrightarrow\exists s_0\in S,s_0(as_1-a_1s)=0$

we may write$\displaystyle (a,s)$ as $\displaystyle \frac{a}{s}$

define the add operation

$\displaystyle \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$

define the times operation

$\displaystyle \frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

Then we can prove that $\displaystyle S^{-1}R$ is a commutative ring of identity.

Prove that $\displaystyle S^{-1}R$has a unique maximal ideal..

Thank you