Editing Field of fractions (section)

=== Localization ===
{{main|Localization (commutative algebra)}}

For any [[commutative ring]] <math>R</math> and any [[multiplicative set]] <math>S</math> in <math>R</math>, the [[localization of a ring|localization]] <math>S^{-1}R</math> is the [[commutative ring]] consisting of [[fraction]]s
:<math>\frac{r}{s}</math>
with <math>r\in R</math> and <math>s\in S</math>, where now <math>(r,s)</math> is equivalent to <math>(r',s')</math> if and only if there exists <math>t\in S</math> such that <math>t(rs'-r's)=0</math>.

Two special cases of this are notable:
* If <math>S</math> is the complement of a [[prime ideal]] <math>P</math>, then <math>S^{-1}R</math> is also denoted <math>R_P</math>.<br/>When <math>R</math> is an [[integral domain]] and <math>P</math> is the zero ideal, <math>R_P</math> is the field of fractions of <math>R</math>.
* If <math>S</math> is the set of non-[[zero-divisor]]s in <math>R</math>, then <math>S^{-1}R</math> is called the [[total quotient ring]].<br/>The [[total quotient ring]] of an [[integral domain]] is its field of fractions, but the [[total quotient ring]] is defined for any [[commutative ring]].

Note that it is permitted for <math>S</math> to contain 0, but in that case <math>S^{-1}R</math> will be the [[trivial ring]].