Editing Localization (commutative algebra) (section)

=== Examples of local properties ===
A property {{mvar|P}} of an {{mvar|R}}-module {{mvar|M}} is a ''local property'' if the following conditions are equivalent:
* {{mvar|P}} holds for {{mvar|M}}.
* {{mvar|P}} holds for all <math>M_\mathfrak{p},</math> where <math>\mathfrak{p}</math> is a prime ideal of {{mvar|R}}.
* {{mvar|P}} holds for all <math>M_\mathfrak{m},</math> where <math>\mathfrak{m}</math> is a maximal ideal of {{mvar|R}}.

The following are local properties:
* {{mvar|M}} is zero.
* {{mvar|M}} is torsion-free (in the case where {{mvar|R}} is a [[commutative domain]]).
* {{mvar|M}} is  a [[flat module]].
* {{mvar|M}} is an [[invertible module]] (in the case where {{mvar|R}} is a commutative domain, and {{mvar|M}} is a submodule of the [[field of fractions]] of {{mvar|R}}).
* <math>f\colon M \to N</math> is injective (resp. surjective), where {{mvar|N}} is another {{mvar|R}}-module.

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.