Editing Localization (commutative algebra) (section)

=== General construction ===
In the general case, a problem arises with [[zero divisor]]s. Let {{mvar|S}} be a multiplicative set in a commutative ring {{mvar|R}}. Suppose that <math>s\in S,</math> and <math>0\ne a\in R</math> is a zero divisor with <math>as=0.</math> Then <math>\tfrac a1</math> is the image in <math>S^{-1}R</math> of <math>a\in R,</math> and one has <math>\tfrac a1 = \tfrac {as}s = \tfrac 0s = \tfrac 01.</math> Thus some nonzero elements of {{mvar|R}} must be zero in <math>S^{-1}R.</math> The construction that follows is designed for taking this into account.

Given {{mvar|R}} and {{mvar|S}} as above, one considers the [[equivalence relation]] on <math>R\times S</math> that is defined by <math>(r_1, s_1) \sim (r_2, s_2)</math> if there exists a <math>t\in S</math> such that <math>t(s_1r_2-s_2r_1)=0.</math>

The localization <math>S^{-1}R</math> is defined as the set of the [[equivalence class]]es for this relation. The class of {{math|(''r'', ''s'')}} is denoted as <math>\frac rs,</math> <math>r/s,</math> or <math>s^{-1}r.</math> So, one has <math>\tfrac{r_1}{s_1}=\tfrac{r_2}{s_2}</math> if and only if there is a <math>t\in S</math> such that <math>t(s_1r_2-s_2r_1)=0.</math> The reason for the <math>t</math> is to handle cases such as the above <math>\tfrac a1 = \tfrac 01,</math> where <math>s_1r_2-s_2r_1</math> is nonzero even though the fractions should be regarded as equal.

The localization <math>S^{-1}R</math> is a commutative ring with addition 
:<math>\frac {r_1}{s_1}+\frac {r_2}{s_2} = \frac{r_1s_2+r_2s_1}{s_1s_2},</math>
multiplication
:<math>\frac {r_1}{s_1}\,\frac {r_2}{s_2} = \frac{r_1r_2}{s_1s_2},</math>
[[additive identity]] <math>\tfrac 01,</math> and [[multiplicative identity]] <math>\tfrac 11.</math>

The [[function (mathematics)|function]] 
:<math>r\mapsto \frac r1</math>
defines a [[ring homomorphism]] from <math>R</math> into <math>S^{-1}R,</math> which is [[injective function|injective]] if and only if {{mvar|S}} does not contain any zero divisors.

If <math>0\in S,</math> then <math>S^{-1}R</math> is the [[zero ring]] that has only one unique element {{math|0}}.

If {{mvar|S}} is the set of all [[zero divisor|regular elements]] of {{mvar|R}} (that is the elements that are not zero divisors), <math>S^{-1}R</math> is called the [[total ring of fractions]] of {{mvar|R}}.