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| In [[abstract algebra]], the '''total quotient ring''',<ref>Matsumura (1980), p. 12</ref> or '''total ring of fractions''',<ref>Matsumura (1989), p. 21</ref> is a construction that generalizes the notion of the [[field of fractions]] of an [[integral domain]] to [[commutative ring]]s ''R'' that may have [[zero divisor]]s. The construction embeds ''R'' in a larger ring, giving every non-zero-divisor of ''R'' an inverse in the larger ring. Nothing more in ''A'' can be given an inverse, if one wants the homomorphism from ''A'' to the new ring to be injective.
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| <!-- The idea is to formally invert as many elements of the ring as possible without making the ring smaller (i. e. without trivializing any nonzero element of the ring).-->
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| == Definition ==
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| Let <math>R</math> be a commutative ring and let <math>S</math> be the set of elements which are not zero divisors in <math>R</math>; then <math>S</math> is a [[multiplicatively closed set]]. Hence we may [[localization of a ring|localize]] the ring <math>R</math> at the set <math>S</math> to obtain the total quotient ring <math>S^{-1}R=Q(R)</math>.
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| If <math>R</math> is a [[integral domain|domain]], then <math>S=R-\{0\}</math> and the total quotient ring is the same as the field of fractions. This justifies the notation <math>Q(R)</math>, which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain.
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| Since <math>S</math> in the construction contains no zero divisors, the natural map <math>R \to Q(R)</math> is injective, so the total quotient ring is an extension of <math>R</math>.
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| == Examples ==
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| The total quotient ring <math>Q(A \times B)</math> of a product ring is the product of total quotient rings <math>Q(A) \times Q(B)</math>. In particular, if ''A'' and ''B'' are integral domains, it is the product of quotient fields.
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| The total quotient ring of the ring of [[holomorphic function]]s on an open set ''D'' of complex numbers is the ring of [[meromorphic function]]s on ''D'', even if ''D'' is not connected.
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| In an [[Artinian ring]], all elements are units or zero divisors. Hence the set of non-zero divisors is the group of units of the ring, <math>R^{\times}</math>, and so <math>Q(R) = (R^{\times})^{-1}R</math>. But since all these elements already have inverses, <math>Q(R) = R</math>.
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| The same thing happens in a commutative [[von Neumann regular ring]] ''R''. Suppose ''a'' in ''R'' is not a zero divisor. Then in a von Neumann regular ring ''a''=''axa'' for some ''x'' in ''R'', giving the equation ''a''(''xa''-1)=0. Since ''a'' is not a zero divisor, ''xa''=1, showing ''a'' is a unit. Here again, <math>Q(R) = R</math>.
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| == Applications ==
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| *The rational functions over a ring ''R''{{dubious|date=December 2013}} can be constructed from the polynomial ring ''R''[''x''] as a total quotient ring.<ref>{{citation|title=Public-key Cryptography: Theory and Practice|first1=Abhijit|last1=Das|first2=C. E. Veni|last2=Madhavan|publisher=Pearson Education India|year=2009|isbn=9788131708323|page=121|url=http://books.google.com/books?id=fzoiOeUf8fIC&pg=PA121}}.</ref>
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| *In [[algebraic geometry]] one considers a [[sheaf (mathematics)|sheaf]] of total quotient rings on a [[scheme (mathematics)|scheme]], and this may be used to give one possible definition of a [[Cartier divisor]].
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| == Generalization ==
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| If <math>R</math> is a commutative ring and <math>S</math> is any [[multiplicatively closed set|multiplicative subset]] in <math>R</math>, the [[Localization of a ring|localization]] <math>S^{-1}R</math> can still be constructed, but the ring homomorphism from <math>R</math> to <math>S^{-1}R</math> might fail to be injective. For example, if <math>0 \in S</math>, then <math>S^{-1}R</math> is the trivial ring.
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| ==Notes==
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| <references/>
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| ==References==
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| *Hideyuki Matsumura, ''Commutative algebra'', 1980
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| *Hideyuki Matsumura, ''Commutative ring theory'', 1989
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| [[Category:Commutative algebra]]
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| [[Category:Ring theory]]
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| [[de:Lokalisierung_(Algebra)#Totalquotientenring]]
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