Localization at primes and local ring

Question

I'm now studying localization. When $R$ is a commutative ring, I know that localization at a prime is a local ring. I have a question.

Let $\mathfrak p_{1},\dots,\mathfrak p_{n}$ be primes of $R$ and set $S=R-\cup_{i=1}^n{\mathfrak p_{i}}$, then is localization $S^{-1}R$ a local ring?

Answer 1

this is 5.35 of the book "Steps in Commutative Algebra" (by R. Y. Sharp). So hint from it:

and

so if there is no inclusion in {$p_{1},...,p_{n}$} then we have $n$ maximal ideals