# localization

## English

### Etymology

From localize +‎ -ation; compare French localisation.

### Noun

localization (countable and uncountable, plural localizations)

1. The act of localizing.
2. (software engineering) The act or process of making a product suitable for use in a particular country or region.
Coordinate terms: internationalization, i18n
3. The state of being localized.
4. (algebra) A systematic method of adding multiplicative inverses to a ring.
5. (algebra) A ring of fractions of a given ring, such that the complement of the set of allowed denominators is an ideal.
• 2007, Ivan Fesenko, “Rings and modules”, in G13ALS Algebra 2, 2007/2008 @ maths.nottingham.ac.uk, page 27:
3) Geometric interpretation of the localization.
Let V be an irreducible algebraic variety. Then P = J(V) is a prime ideal of $\mathbb {C} [X_{1},...,X_{n}]$ and so $\mathbb {C} [V]=\mathbb {C} [X_{1},...,X_{n}]/J(V)$ is an integral domain.
The localization $\mathbb {C} [X_{1},...,X_{n}]_{P}$ is a subring of $\mathbb {C} (X_{1},...,X_{n})$ consisting of rational functions $\{f/g:f,g\in \mathbb {C} [X_{1},...,X_{n}],g\notin P\}$ which are defined on a nonempty subset of V. If V = {x} is a point, then P is maximal and $\mathbb {C} [X_{1},...,X_{n}]_{P}=\{f/g:f,g\in \mathbb {C} [X_{1},...,X_{n}],g(x)\neq 0\}$ consists of rational functions which are defined at x.

#### Translations

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