Editing Localization (commutative algebra) (section)

=== Examples ===
*If <math>R=\Z</math> is the ring of [[integer]]s, and <math>S=\Z\setminus \{0\},</math> then <math>S^{-1}R</math> is the field <math>\Q</math> of the [[rational number]]s.
*If {{mvar|R}} is an [[integral domain]], and <math>S=R\setminus \{0\},</math> then <math>S^{-1}R</math> is the [[field of fractions]] of {{mvar|R}}. The preceding example is a special case of this one.
*If {{mvar|R}} is a [[commutative ring]], and if {{mvar|S}} is the subset of its elements that are not [[zero divisor]]s, then <math>S^{-1}R</math> is the [[total ring of fractions]] of {{mvar|R}}. In this case, {{mvar|S}} is the largest multiplicative set such that the homomorphism <math>R\to S^{-1}R</math> is injective. The preceding example is a special case of this one.
*If <math>x</math> is an element of a commutative ring {{mvar|R}} and <math>S=\{1, x, x^2, \ldots\},</math> then <math>S^{-1}R</math> can be identified (is [[canonical isomorphism|canonically isomorphic]] to) <math>R[x^{-1}]=R[s]/(xs-1).</math> (The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an [[affine scheme]].
*If <math>\mathfrak p</math> is a [[prime ideal]] of a commutative ring {{mvar|R}}, the [[set complement]] <math>S=R\setminus \mathfrak p</math> of <math>\mathfrak p</math> in {{mvar|R}} is a multiplicative set (by the definition of a prime ideal). The ring <math>S^{-1}R</math> is a [[local ring]] that is generally denoted <math>R_\mathfrak p,</math> and called ''the local ring of {{mvar|R}} at'' <math>\mathfrak p.</math> This sort of localization is fundamental in [[commutative algebra]], because many properties of a commutative ring can be read on its local rings. Such a property is often called a [[local property]]. For example, a ring is [[regular ring|regular]] if and only if all its local rings are regular.