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| {{for|Ore's condition in graph theory|Ore's theorem}}
| | Andrew Simcox is the name his parents gave him and he completely loves this name. The favorite hobby for him and his children is to play lacross and he would never give it up. Ohio is exactly where her house is. Invoicing is my profession.<br><br>Here is my web-site: [http://Medialab.Zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- tarot card readings] |
| {{More footnotes|date=April 2012}}
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| In [[mathematics]], especially in the area of [[algebra]] known as [[ring theory]], the '''Ore condition''' is a condition introduced by [[Øystein Ore]], in connection with the question of extending beyond [[commutative ring]]s the construction of a [[field of fractions]], or more generally [[localization of a ring]]. The ''right Ore condition'' for a [[multiplicative subset]] ''S'' of a [[ring (mathematics)|ring]] ''R'' is that for ''a'' ∈ ''R'' and ''s'' ∈ ''S'', the intersection ''aS'' ∩ ''sR'' ≠ {{unicode|∅}}.<ref>{{cite book|last=Cohn|first=P. M.|title=Algebra |volume=Vol. 3 |edition=2nd |year=1991|chapter=Chap. 9.1| pages=351}}</ref> A domain that satisfies the right Ore condition is called a '''right Ore domain'''. The left case is defined similarly.
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| ==General idea==
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| The goal is to construct the right ring of fractions ''R''[''S''<sup>−1</sup>] with respect to [[multiplicative subset]] ''S''. In other words we want to work with elements of the form ''as''<sup>−1</sup> and have a ring structure on the set ''R''[''S''<sup>−1</sup>]. The problem is that there is no obvious interpretation of the product (''as''<sup>−1</sup>)(''bt''<sup>−1</sup>); indeed, we need a method to "move" ''s''<sup>−1</sup> past ''b''. This means that we need to be able to rewrite ''s''<sup>−1</sup>''b'' as a product ''b''<sub>1</sub>''s''<sub>1</sub><sup>−1</sup>.<ref>{{cite web|last=Artin|first=Michael|title=Noncommutative Rings|url=http://math.mit.edu/~etingof/artinnotes.pdf|accessdate=9 May 2012|pages=13|year=1999}}</ref> Suppose ''s''<sup>−1</sup>''b'' = ''b''<sub>1</sub>''s''<sub>1</sub><sup>−1</sup> then multiplying on the left by ''s'' and on the right by ''s''<sub>1</sub>, we get ''bs''<sub>1</sub> = ''sb''<sub>1</sub>. Hence we see the necessity, for a given ''a'' and ''s'', of the existence of ''a''<sub>1</sub> and ''s''<sub>1</sub> with ''s''<sub>1</sub> ≠ 0 and such that ''as''<sub>1</sub> = ''sa''<sub>1</sub>.
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| ==Application==
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| Since it is well known that each [[integral domain]] is a subring of a field of fractions (via an embedding) in such a way that every element is of the form ''rs''<sup>−1</sup> with ''s'' nonzero, it is natural to ask if the same construction can take a noncommutative domain and associate a [[division ring]] (a noncommutative field) with the same property. It turns out that the answer is sometimes "no", that is, there are domains which do not have an analogous "right division ring of fractions".
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| For every right Ore domain ''R'', there is a unique (up to natural ''R''-isomorphism) division ring ''D'' containing ''R'' as a subring such that every element of ''D'' is of the form ''rs''<sup>−1</sup> for ''r'' in ''R'' and ''s'' nonzero in ''R''. Such a division ring ''D'' is called a '''ring of right fractions''' of ''R'', and ''R'' is called a '''right order''' in ''D''. The notion of a '''ring of left fractions''' and '''left order''' are defined analogously, with elements of ''D'' being of the form ''s''<sup>−1</sup>''r''.
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| It is important to remember that the definition of ''R'' being a right order in ''D'' includes the condition that ''D'' must consist entirely of elements of the form ''rs''<sup>−1</sup>. Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean ''R'' is a left order in ''D'', since it is possible ''D'' has an element which is not of the form ''s''<sup>−1</sup>''r''. Thus it is possible for ''R'' to be a right-not-left Ore domain. Intuitively, the condition that all elements of ''D'' be of the form ''rs''<sup>−1</sup> says that ''R'' is a "big" ''R''-submodule of ''D''. In fact the condition ensures ''R''<sub>''R''</sub> is an [[essential submodule]] of ''D''<sub>''R''</sub>. Lastly, there is even an example of a domain in a division ring which satisfies ''neither'' Ore condition (see examples below).
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| Another natural question is: "When is a subring of a division ring right Ore?" One characterization is that a subring ''R'' of a division ring ''D'' is a right Ore domain if and only if ''D'' is a [[flat module|flat]] left ''R''-module {{harv|Lam|2007|loc=Ex. 10.20}}.
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| A different, stronger version of the Ore conditions is usually given for the case where ''R'' is not a domain, namely that there should be a common multiple
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| :''c'' = ''au'' = ''bv''
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| with ''u'', ''v'' not [[zero divisor]]s. In this case, '''Ore's theorem''' guarantees the existence of an [[over-ring]] called the (right or left) '''classical ring of quotients'''.
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| ==Examples==
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| Commutative domains are automatically Ore domains, since for nonzero ''a'' and ''b'', ''ab'' is nonzero in ''aR''∩''bR''. Right [[Noetherian ring|Noetherian]] domains, such as right [[principal ideal domain]]s, are also known to be right Ore domains. Even more generally, [[Alfred Goldie]] proved that a domain ''R'' is right Ore if and only if ''R''<sub>''R''</sub> has finite [[uniform dimension]]. It is also true that right [[Bézout domain]]s are right Ore.
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| A subdomain of a division ring which is not right or left Ore: If ''F'' is any field, and <math>G = \langle x,y \rangle\,</math> is the [[free monoid]] on two symbols ''x'' and ''y'', then the [[monoid ring]] <math>F[G]\,</math> does not satisfy any Ore condition, but it is a [[free ideal ring]] and thus indeed a subring of a division ring, by {{harv|Cohn|1995|loc=Cor 4.5.9}}.
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| ==Multiplicative sets==
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| The Ore condition can be generalized to other [[multiplicative subset]]s, and is presented in textbook form in {{harv|Lam|1999|loc=§10}} and {{harv|Lam|2007|loc=§10}}. A subset ''S'' of a ring ''R'' is called a '''right denominator set''' if it satisfies the following three conditions for every ''a'',''b'' in ''R'', and ''s'', ''t'' in ''S'':
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| # ''st'' in ''S''; (The set ''S'' is '''multiplicatively closed'''.)
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| # ''aS'' ∩ ''sR'' is not empty; (The set ''S'' is '''right permutable'''.)
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| # If ''sa'' = 0, then there is some ''u'' in ''S'' with ''au'' = 0; (The set ''S'' is '''right reversible'''.)
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| If ''S'' is a right denominator set, then one can construct the '''ring of right fractions''' ''RS''<sup>−1</sup> similarly to the commutative case. If ''S'' is taken to be the set of regular elements (those elements ''a'' in ''R'' such that if ''b'' in ''R'' is nonzero, then ''ab'' and ''ba'' are nonzero), then the right Ore condition is simply the requirement that ''S'' be a right denominator set.
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| Many properties of commutative localization hold in this more general setting. If ''S'' is a right denominator set for a ring ''R'', then the left ''R''-module ''RS''<sup>−1</sup> is [[flat module|flat]]. Furthermore, if ''M'' is a right ''R''-module, then the ''S''-torsion, tor<sub>''S''</sub>(''M'') = { ''m'' in ''M'' : ''ms'' = 0 for some ''s'' in ''S'' }, is an ''R''-submodule isomorphic to Tor<sub>1</sub>(''M'',''RS''<sup>−1</sup>), and the module ''M'' ⊗<sub>''R''</sub> ''RS''<sup>−1</sup> is naturally isomorphic to a module ''MS''<sup>−1</sup> consisting of "fractions" as in the commutative case.
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| ==Notes==
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| <references />
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| ==References==
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| * {{cite book|last=Cohn|first=P. M.|title=Algebra |volume=Vol. 3 |edition=2nd |year=1991|publisher=John Wiley & Sons Ltd|location=Chichester|pages=xii+474|chapter= 9.1|isbn=0-471-92840-2 | id={{MathSciNet | id = 1098018}} }}
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| *{{Citation | last=Cohn | first=P.M. | title=On the embedding of rings in skew fields | journal=Proc. London Math. Soc. | volume=11 | year=1961 | pages=511–530 | doi=10.1112/plms/s3-11.1.511 | mr=25#100 }}
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| * Cohn, P.M. (1995) ''Skew fields, Theory of general division rings'', [[Cambridge University Press]], ISBN 0-521-43217-0
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | year=1999}}
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| *{{Citation | last1=Lam | first1=Tsit-Yuen | title=Exercises in modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Problem Books in Mathematics | isbn=978-0-387-98850-4; 978-0-387-98850-4 | id={{MathSciNet | id = 2278849}} | year=2007}}
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| *{{citation|author=Stenström, Bo |title=Rings and modules of quotients |series=Lecture Notes in Mathematics, Vol. 237 |publisher=Springer-Verlag |place=Berlin |date=1971 |pages=vii+136 |isbn=978-3-540-05690-4| id={{MathSciNet | id = 0325663}} }}
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| ==External links==
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| * [http://planetmath.org/encyclopedia/OreCondition.html PlanetMath page on Ore condition]
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| * [http://planetmath.org/?op=getobj&from=objects&name=OresTheorem2 PlanetMath page on Ore's theorem]
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| * [http://planetmath.org/encyclopedia/ClassicalRingOfQuotients.html PlanetMath page on classical ring of quotients]
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| [[Category:Ring theory]]
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Andrew Simcox is the name his parents gave him and he completely loves this name. The favorite hobby for him and his children is to play lacross and he would never give it up. Ohio is exactly where her house is. Invoicing is my profession.
Here is my web-site: tarot card readings