An operation on rings and modules. Given a commutative unit ring , and a subset of , closed under multiplication, such that , and , the localization of at is the ring
(1)
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where the addition and the multiplication of the formal fractions are defined according to the natural rules,
(2)
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and
(3)
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The ring is a subring of via the identification .
For an -module , the localization of at is defined as the tensor product , i.e., as the set of linear combinations of the elementary tensors
(4)
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which are also denoted for short.
The properties required for the subset are fulfilled by
1. The set of non zero-divisors of ; in this case is the ring of fractions of .
2. The complement of any prime ideal of : in this case the clumsy notation is replaced by . This ring is called the localization of at , and it is a local ring, with maximal ideal .
The name given to this operation derives from the geometric meaning it takes when applied to the rings associated with algebraic varieties.
The union of the coordinate axes of the real Cartesian plane is an algebraic variety given by the equation
(5)
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and is associated with the quotient ring . The localization of at the maximal ideal generated by the residues of and of , denoted , describes at the origin; its algebraic properties provide clues to local geometric properties. For example, the localized ring is nonregular since its Krull dimension is 1, whereas two elements are needed to generate its maximal ideal . This is the algebraic counterpart of the fact that the origin is a singular point (a knot) for . For all other points of the -axis, , the localized ring is regular of dimension 1, since generates the whole maximal ideal : being , one has that . The same argument applies to the -axis. It follows that outside the origin variety is regular in the geometric meaning of "smooth" or "nonsingular."