Example 10.9.8. Let $A$ be a ring and let $M$ be an $A$-module. Here are some important examples of localizations.

1. Given $\mathfrak p$ a prime ideal of $A$ consider $S = A\setminus \mathfrak p$. It is immediately checked that $S$ is a multiplicative set. In this case we denote $A_\mathfrak p$ and $M_\mathfrak p$ the localization of $A$ and $M$ with respect to $S$ respectively. These are called the localization of $A$, resp. $M$ at $\mathfrak p$.

2. Let $f\in A$. Consider $S = \{ 1, f, f^2, \ldots \}$. This is clearly a multiplicative subset of $A$. In this case we denote $A_ f$ (resp. $M_ f$) the localization $S^{-1}A$ (resp. $S^{-1}M$). This is called the localization of $A$, resp. $M$ with respect to $f$. Note that $A_ f = 0$ if and only if $f$ is nilpotent in $A$.

3. Let $S = \{ f \in A \mid f \text{ is not a zerodivisor in }A\}$. This is a multiplicative subset of $A$. In this case the ring $Q(A) = S^{-1}A$ is called either the total quotient ring, or the total ring of fractions of $A$.

4. If $A$ is a domain, then the total quotient ring $Q(A)$ is the field of fractions of $A$. Please see Fields, Example 9.3.4.

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