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| : ''"Quotient field" redirects here. It should not be confused with a [[quotient ring]].''
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| In [[abstract algebra]], the '''field of fractions''' of an [[integral domain]] is the smallest [[field (mathematics)|field]] in which it can be embedded. The elements of the field of fractions of the integral domain ''R'' have the form ''a/b'' with ''a'' and ''b'' in ''R'' and ''b'' ≠ 0. The field of fractions of ''R'' is sometimes denoted by Quot(''R'') or Frac(''R'').
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| Mathematicians refer to this construction as the ''field of fractions'', ''fraction field'', ''field of quotients'', or ''quotient field''. All three are in common usage. The expression "quotient field" may sometimes run the risk of confusion with the quotient of a ring by an ideal, which is a quite different concept.
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| A multiplicative identity is not required for the role of the integral domain; this construction can be applied to any [[zero ring|nonzero]] [[commutativity|commutative]] [[pseudo-ring]] with no nonzero [[zero divisor]]s.<ref>Rings, Modules, and Linear Algebra: Hartley, B & Hawkes, T.O. 1970</ref>
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| == Examples ==
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| * The field of fractions of the ring of [[integer]]s is the field of [[rational number|rationals]], '''Q''' = Quot('''Z''').
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| * Let ''R'' := { ''a'' + ''b'' i | ''a'',''b'' in '''Z''' } be the ring of [[Gaussian integer]]s. Then Quot(''R'') = {''c'' + ''d'' i | ''c'',''d'' in '''Q'''}, the field of [[Gaussian rational]]s.
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| * The field of fractions of a field is canonically [[isomorphism|isomorphic]] to the field itself.
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| * Given a field ''K'', the field of fractions of the [[polynomial ring]] in one indeterminate ''K''[''X''] (which is an integral domain), is called the '''{{visible anchor|field of rational functions}}''' or '''field of rational fractions'''<ref>{{cite book|author=Ėrnest Borisovich Vinberg|title=A course in algebra|page=131|year=2003|url=http://books.google.com/books?id=rzNq39lvNt0C&pg=PA132&dq=%22rational+fraction%22&hl=fr&ei=JiucTp-qJIj0sgbY2PSfBA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CEIQ6AEwBDgK}}</ref><ref>{{cite book|author=Stephan Foldes|title=Fundamental structures of algebra and discrete mathematics|page=128|year=1994|url=http://books.google.com/books?id=IR-rH0vLyz0C&pg=PA128&dq=%22field+of+rational+fractions%22&hl=fr&ei=z2KcTpmTNY3Tsgaog82WBA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwADgK#v=onepage&q=%22field%20of%20rational%20fractions%22&f=false}}</ref><ref>{{cite book|author=Pierre Antoine Grillet|title=Abstract algebra|page=124|year=2007|url=http://books.google.com/books?id=LJtyhu8-xYwC&pg=PA124&dq=%22field+of+rational+fractions%22&hl=fr&ei=42GcTveIJc7JswaP1LGMBA&sa=X&oi=book_result&ct=result&resnum=5&ved=0CD8Q6AEwBA#v=onepage&q=%22field%20of%20rational%20fractions%22&f=false}}</ref> and is denoted ''K''(''X'').
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| == Construction ==
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| Let ''R'' be any nonzero commutative pseudo-ring with no nonzero [[zero divisor]]s.
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| For ''n'', ''d'' ∈ ''R'' with ''d'' ≠ 0, the fraction ''n''/''d'' denotes the equivalence class of pairs (''n'',''d''), where (''n'',''d'') is equivalent to (''m'',''b'') [[if and only if]] ''nb'' = ''md''.
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| (The definition of equivalence is modeled on the property of rational numbers that ''n''/''d'' = ''m''/''b'' if and only if ''nb'' = ''md''.)
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| The ''field of fractions'' Quot(''R'') is defined as the set of all such fractions ''n''/''d''.
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| The sum of ''n''/''d'' and ''m''/''b'' is defined as <math>\frac{nb+md}{db}</math>, and the product of ''n''/''d'' and ''m''/''b'' is defined as <math>\frac{nm}{db}</math> (one checks that these are well defined).
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| The embedding of ''R'' in Quot(''R'') maps each ''n'' in ''R'' to the fraction <math>\frac{en}{e}</math> for any nonzero ''e'' in ''R'' (the equivalence class is independent of the choice of ''e''). This is modeled on the identity ''n''/1 = ''n''. If additionally, ''R'' contains a [[multiplicative identity]] (that is, ''R'' is an [[integral domain]]), then <math>\frac{en}{e} = \frac{n}{1}</math>.
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| The field of fractions of ''R'' is characterized by the following [[universal property]]: if ''h'': ''R'' → ''F'' is an injective [[ring homomorphism]] from ''R'' into a field ''F'', then there exists a unique ring homomorphism ''g'' : Quot(''R'') → ''F'' which extends ''h''.
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| There is a [[category theory|categorical]] interpretation of this construction. Let '''C''' be the category of integral domains and [[injective]] ring maps. The [[functor]] from '''C''' to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the [[adjoint functor|left adjoint]] of the [[forgetful functor]] from the category of fields to '''C'''.
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| == Generalisation ==
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| {{main|Localization of a ring}}
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| For any commutative ring ''R'' and any [[multiplicative set]] ''S'' in ''R'', the [[localization of a ring|localization]] ''S''<sup>−1</sup>''R'' is the commutative ring consisting of fractions ''r''/''s'' with ''r'' ∈ ''R'' and ''s'' ∈ ''S'',
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| where now (''r'',''s'') is equivalent to (''r''′,''s''′) if and only if there exists ''t'' ∈ ''S'' such that ''t''(''rs''′-''r''′''s'') = 0.
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| Two special cases of this are notable:
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| * If ''S'' is the complement of a prime ideal ''P'', then ''S''<sup>−1</sup>''R'' is also denoted ''R''<sub>''P''</sub>. When ''R'' is an integral domain and ''P'' is the zero ideal, ''R''<sub>''P''</sub> is the field of fractions of ''R''.
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| * If ''S'' is the set of non-zero-divisors in ''R'', then ''S''<sup>−1</sup>''R'' is called the [[total quotient ring]]. The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring.
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| == See also ==
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| * [[Ore condition]]; this is the condition one needs to consider in the noncommutative case.
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| == References ==
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| {{reflist}}
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| [[Category:Field theory]]
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| [[Category:Commutative algebra]]
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