Editing Localization (commutative algebra) (section)

===Multiplicative set===
Localization is commonly done with respect to a [[multiplicatively closed set]] {{mvar|S}} (also called a ''multiplicative set'' or a ''multiplicative system'') of elements of a ring {{mvar|R}}, that is a subset of {{mvar|R}} that is [[closure (mathematics)|closed]] under multiplication, and contains {{math|1}}.

The requirement that {{mvar|S}} must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to {{mvar|S}}. The localization by a set {{mvar|U}} that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of {{mvar|U}}. However, the same localization is obtained by using the multiplicatively closed set {{mvar|S}} of all products of elements of {{mvar|U}}. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.

For example, the localization by a single element {{mvar|s}} introduces fractions of the form <math>\tfrac a s,</math> but also products of such fractions, such as  <math>\tfrac {ab} {s^2}.</math> So, the denominators will belong to the multiplicative set <math>\{1, s, s^2, s^3,\ldots\}</math> of the powers of {{mvar|s}}. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".

The localization of a ring {{mvar|R}} by a multiplicative set {{mvar|S}} is generally denoted <math>S^{-1}R,</math> but other notations are commonly used in some special cases: if <math>S= \{1, t, t^2,\ldots \}</math> consists of the powers of a single element, <math>S^{-1}R</math> is often denoted <math>R_t;</math> if <math>S=R\setminus \mathfrak p</math> is the [[complement (set theory)|complement]] of a [[prime ideal]] <math>\mathfrak p</math>, then <math>S^{-1}R</math> is denoted <math>R_\mathfrak p.</math>

''In the remainder of this article, only localizations by a multiplicative set are considered.''