Editing Localization (commutative algebra) (section)

=== Ring properties ===
Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning [[ideal (ring theory)|ideals]], [[module (mathematics)|modules]], or several multiplicative sets are considered in other sections.

* <math>S^{-1}R = 0</math> [[if and only if]] {{math|''S''}} contains {{math|0}}.
* The [[ring homomorphism]] <math>R\to S^{-1}R</math> is injective if and only if {{math|''S''}} does not contain any [[zero divisor]]s.
* The ring homomorphism <math>R\to S^{-1}R</math> is an [[epimorphism]] in the [[category of rings]], that is not [[surjective]] in general.
* The ring <math>S^{-1}R</math> is a [[flat module|flat {{mvar|R}}-module]] (see {{slink||Localization of  a module}} for details).
* If <math>S=R\setminus \mathfrak p</math> is the [[complement (set theory)|complement]] of a prime ideal <math>\mathfrak p</math>, then <math>S^{-1} R,</math> denoted <math>R_\mathfrak p,</math> is a [[local ring]]; that is, it has only one [[maximal ideal]].
<!--Properties to be moved in another section-->
*Localization commutes with formations of finite sums, products, intersections and radicals;<ref>{{harvnb|Atiyah|Macdonald|1969|loc=Proposition 3.11. (v).}}</ref> e.g., if <math>\sqrt{I}</math> denote the [[radical of an ideal]] ''I'' in ''R'', then
::<math>\sqrt{I} \cdot S^{-1}R = \sqrt{I \cdot S^{-1}R}\,.</math>
:In particular, ''R'' is [[reduced ring|reduced]] if and only if its total ring of fractions is reduced.<ref>Borel, AG. 3.3</ref>
*Let ''R'' be an integral domain with the field of fractions ''K''. Then its localization <math>R_\mathfrak{p}</math> at a prime ideal <math>\mathfrak{p}</math> can be viewed as a subring of ''K''. Moreover,
::<math>R = \bigcap_\mathfrak{p} R_\mathfrak{p} = \bigcap_\mathfrak{m} R_\mathfrak{m}</math>
:where the first intersection is over all prime ideals and the second over the maximal ideals.<ref>Matsumura, Theorem 4.7</ref>
* There is a [[bijection]] between the set of prime ideals of ''S''<sup>&minus;1</sup>''R'' and the set of prime ideals of ''R'' that are [[Disjoint sets|disjoint]] from ''S''. This bijection is induced by the given homomorphism ''R'' → ''S''<sup>&nbsp;&minus;1</sup>''R''.