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For example the ring [[modular arithmetic|'''Z'''/''n'''''Z''']] where ''n'' is [[composite number|composite]] is not an integral ___domain. When ''n'' is a [[prime number|prime]] power it is a finite [[local ring]], and its elements are either units or [[nilpotent]]. This implies it can be localized only to a zero ring. But when ''n'' can be factorised as ''ab'' with ''a'' and ''b'' [[coprime]] and greater than 1, then '''Z'''/''n'''''Z''' is by the [[Chinese remainder theorem]] isomorphic to '''Z'''/''a'''''Z''' × '''Z'''/''b'''''Z'''. If we take ''S'' to consist only of (1,0) and 1 = (1,1), then the corresponding localization is '''Z'''/''a'''''Z'''.
Two classes of localizations occur commonly in [[commutative algebra]] and [[algebraic geometry]]:
* The set ''S'' consists of all powers of a given element ''r''. In algebraic geometry, these localizations are used to construct the rings of functions on [[open set|open subsets]] in [[Zariski topology]] of the [[spectrum of a ring]], Spec(''R''). For example, if ''R'' = ''K''[''X''] is the [[polynomial ring]] and ''r'' = ''X'' then the localization produces the ring of [[Laurent polynomial]]s ''K''[''X'', ''X''<sup>−1</sup>]. In this case, localization corresponds to the embedding ''U'' ⊂ ''A''<sup>1</sup>, where ''A''<sup>1</sup> is the affine line and ''U'' is its Zariski open subset which is the complement of 0.
* ''S'' is the [[complement_(set theory)|complement]] of a given [[prime ideal]] ''P'' in ''R'' (this is a multiplicatively closed set). In this case, one also speaks of the "localization at ''P''".
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