On the other hand, some properties are not local properties. For example, an infinite [[direct product]] of [[field (mathematics)|fields]] is not an [[integral ___domain]] nor a [[Noetherian ring]], while all its local rings are fields, and therefore Noetherian integral domains.
== Localization to Zariski open sets ==
Let <math>R</math> be a commutative ring. We can define its [[Spectrum of a ring|spectrum]] <math>\operatorname{Spec}(R)</math> to be a [[ringed space]] whose:
* ''Points'' are the prime ideals of <math>R</math>,
* c''losed sets'' correspond in some way (which we shall describe below) to the [[ideals]] of <math>R</math>,
* and which has a [[sheaf of rings]] <math>\mathcal O</math> attached to it.
A [[Zariski topology|Zariski closed set]] <math>U^C</math>of <math>\operatorname{Spec}(R)</math> corresponds in a certain way to an ideal <math>I \subset R</math>. More precisely, <math>U^C</math> consists of the set of prime ideals <math>P \subset R</math> which are supersets of <math>I</math>. Therefore <math>U</math>, which is Zariski ''open'', consists of the prime ideals <math>P</math> which are not supersets of <math>I</math>.
The ring <math>\mathcal O(U)</math> is then defined to be the localisation of <math>R</math> by the multiplicative set <math>\{f \in R: \forall \text{maximal ideals } M \text{ not containing }I, f \not \in M\}</math>.
