Definition of localization of rings

Question

I'm trying to understand this definition of Hungerford's book:

The definition is simple, I think I understood what the author means, but...

What is $P_P$? because we will have $P_P=S^{-1}P$, with $S=P-P=\varnothing$.

What is $S^{^-1}P$, with $S=\varnothing$?

I'm sure it should be a silly think

I need help.

Thanks in advance.

Answer 1

No: $P_P$ is $S^{-1}P$ where still $S=R\setminus P$. In other words, it is the ideal generated by (the elements in) $P$ in the localized ring $R_P$.

Answer 2

$S$ is not $P \backslash P$ - it's still $R \backslash P$.

Look at an example: consider the localization $\mathbb{Z}_{(p)} = \{\frac{a}{b} : p \nmid b\, \; a,b \in \mathbb{Z}\} \subseteq \mathbb{Q}$ of $\mathbb{Z}$. The unique maximal ideal of this ring is $$p \mathbb{Z}_{(p)} = \{\frac{p \cdot a}{b} : p \nmid b, \; a,b \in \mathbb{Z}\}.$$