Editing Localization (commutative algebra) (section)

=== Integral domains ===
When the ring {{mvar|R}} is an [[integral domain]] and {{mvar|S}} does not contain {{math|0}}, the ring <math>S^{-1}R</math> is a subring of the [[field of fractions]] of {{mvar|R}}. As such, the localization of a domain is a domain.

More precisely, it is the [[subring]] of the field of fractions of {{mvar|R}}, that consists of the fractions <math>\tfrac a s</math> such that <math>s\in S.</math> This is a subring since the sum <math>\tfrac as + \tfrac bt = \tfrac {at+bs}{st},</math> and the product <math>\tfrac as \, \tfrac bt = \tfrac {ab}{st}</math> of two elements of <math>S^{-1}R</math> are in <math>S^{-1}R.</math> This results from the defining property of a multiplicative set, which implies also that <math>1=\tfrac 11\in S^{-1}R.</math> In this case, {{mvar|R}} is a subring of <math>S^{-1}R.</math> It is shown below that this is no longer true in general, typically when {{mvar|S}} contains [[zero divisor]]s.

For example, the [[decimal fraction]]s are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, <math>S^{-1}R</math> consists of the rational numbers that can be written as <math>\tfrac n{10^k},</math> where {{mvar|n}} is an integer, and {{mvar|k}} is a nonnegative integer.