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==Localization at primes==
The definition of a [[prime ideal]] implies immediately that the [[set complement|complement]] <math>S=R\setminus \mathfrak p</math> of a prime ideal <math>\mathfrak p</math> in a commutative ring {{mvar|R}} is a multiplicative set. In this case, the localization <math>S^{-1}R</math> is commonly denoted <math>R_\mathfrak p.</math> The ring <math>R_\mathfrak p</math> is a [[local ring]], that is called ''the local ring of {{mvar|R}}'' at <math>\mathfrak p.</math> This means that <math>\mathfrak p\,R_\mathfrak p=\mathfrak p\otimes_R R_\mathfrak p</math> is the unique [[maximal ideal]] of the ring <math>R_\mathfrak p.</math> Analogously one can define the localization of a module {{mvar|M}} at a prime ideal <math>\mathfrak p</math> of {{mvar|R}}. Again, the localization <math>S^{-1}M</math> is commonly denoted <math>M_{\mathfrak p}</math>.
Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of [[Nakayama lemma]]. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is [[regular ring|regular]] if and only if all its local rings are [[regular local ring]]s.
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