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

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 $\mathfrak p$*.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 $A$.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 $A$.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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