Editing Localization (commutative algebra) (section)

== Terminology explained by the context ==
The term ''localization'' originates in the general trend of modern mathematics to study [[geometry|geometrical]] and [[topology|topological]] objects ''locally'', that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of [[manifold]]s, [[germ (mathematics)|germs]] and [[sheaf (mathematics)|sheafs]]. In [[algebraic geometry]], an [[affine algebraic set]] can be identified with a [[quotient ring]] of a [[polynomial ring]] in such a way that the points of the algebraic set correspond to the [[maximal ideal]]s of the ring (this is [[Hilbert's Nullstellensatz]]). This correspondence has been generalized for making the set of the [[prime ideal]]s of a [[commutative ring]] a [[topological space]] equipped with the [[Zariski topology]]; this topological space is called the [[spectrum of a ring|spectrum of the ring]].

In this context, a ''localization'' by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as ''points'') that do not intersect the multiplicative set.

Two classes of localizations are more commonly considered:
* The multiplicative set is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math> of a ring {{mvar|R}}. In this case, one  speaks of the "localization at <math>\mathfrak p</math>", or "localization at a point". The resulting ring, denoted <math>R_\mathfrak p</math> is a [[local ring]], and is the algebraic analog of a [[germ (mathematics)#Ring of germs|ring of germs]].
* The multiplicative set consists of all powers of an element {{mvar|t}} of a ring {{mvar|R}}. The resulting ring is commonly denoted <math>R_t,</math> and its spectrum is the Zariski open set of the prime ideals that do not contain {{mvar|t}}. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a [[neighborhood basis]] consisting of Zariski open sets of this form).

{{anchor|away from}}In [[number theory]] and [[algebraic topology]], when working over the ring <math>\Z</math> of [[integer]]s, one refers to a property relative to an integer {{mvar|n}} as a property true ''at'' {{mvar|n}} or ''away'' from {{mvar|n}}, depending on the localization that is considered. "'''Away from''' {{mvar|n}}" means that the property is considered after localization by the powers of {{mvar|n}}, and, if {{mvar|p}} is a [[prime number]], "at {{mvar|p}}" means that the property is considered after localization at the prime ideal <math>p\Z</math>. This terminology can be explained by the fact that, if {{mvar|p}} is prime, the nonzero prime ideals of the localization of <math>\Z</math> are either the [[singleton set]] {{math|{{mset|p}}}} or its complement in the set of prime numbers.