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| In [[mathematics]], a '''real closed ring''' is a [[commutative ring]] ''A'' that
| | Buying absinthe online is significantly less easy as walking right into a liquor store and getting a bottle of absinthe. Although a little difficult, buying absinthe on the internet is still possible and you've got to have few precautions. If you have any kind of inquiries pertaining to where and how to use [http://www.Google.com/ get The facts], you could call us at our own website. US citizens face difficulty when they have got to buy absinthe online, as thujone that's an essential active ingredient in absinthe is forbidden in the United States. However, it's been demonstrated that thujone doesn't cause hallucinations similar to cannabis as earlier considered.<br><br> |
| is a subring of a product of [[real closed field]]s, which is closed under
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| continuous [[Semialgebraic set|semi-algebraic]] functions defined over the integers.
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| == Examples of real closed rings == | | [http://www.youtube.com/watch?v=07tJEWdqSr8 youtube.com]Production and sale of absinthe in the United States is strictly prohibited; however, importing absinthe that does not consist of any traces of thujone is permitted. For purchasing any type of alcohol you should be 21 years or older and anyone found violating this law is punished. There are many who are thinking about buying absinthe in the United States. So what is it that pulls lovers of fine liquor to absinthe?<br><br>Well, absinthe is regarded as the famous and controversial liquor made from herbs like wormwood (Artemisia absinthium), green anise, fennel seed, angelica root, star anise hyssop, melissa, and peppermint leaves. The herb wormwood is the most important ingredient and it consists of thujone that's answerable for absinthe unique effects. Absinthe liquor is bitter to taste thus it's always taken with sugar and ice-cold water. Absinthe drink is geared up using a traditional ritual.<br><br>In the traditional French ritual a special absinthe glass, absinthe spoon, sugar cube, and ice-cold water is needed. An ounce or two of absinthe is put in the glass and an absinthe spoon is kept over the rim of the glass. The absinthe spoon has perforations in its basin. A sugar cube click this link is kept on the spoon and ice-cold water is poured over the cube from an absinthe fountain, as the water melts the sugar cube it falls through the perforations into the glass. The clear green absinthe becomes turbid or milky white for the reason that essential oils from your herbs get precipitated. This is whats called louching of absinthe. Some more ice-cold water is added and the drink is prepared. Absinthe was extremely popular in Europe in the 18th and nineteenth century and was adoringly known as "The Green Fairy" or "Green Muse". However, due to specific misconceptions and rumors Visit us at absinthe was prohibited in Europe and North America during the 20th century.<br><br>Within the light of new evidence introduced by scientists in the year 1999 most European countries removed the ban on absinthe and it is now accessible to people. In the United States it is still prohibited; however, US citizens can buy absinthe online from a variety of online stores. Absinthe is a powerful alcoholic drink and the alcohol content is as high as 60%.<br><br>Since buying real absinthe is against the law in the US many replicas of absinthe can be bought online. You ought to be cautious with such fraudulent websites which promise to ship you the real thing. You can buy absinthe online from a number of the most respected distillers in Europe. The absinthe shipped will be minus thujone and perfectly legal. Ordering absinthe from French distilleries is also a wise decision and the product you get does not consist of thujone. Even so, when ordering absinthe online you need to be cautious as some websites may be illegal.<br><br>Possibly the most suitable option intended for US citizens is to buy absinthe kits from non-US producers and ready your own absinthe in your own home. The kit includes all of the essential herbs, recommendations on preparation, rituals along with other accessories just like spoon, straining cloth, etc you can prepare about 2 liters of absinthe liquor from one kit that will keep a party of eight going until dawn. Serving absinthe at a party may also enable you to get the title "King of good times". One place where you can be assured of high quality herbs and genuine absinthe essence is absinthekit.com<br><br>You should order your kit and spend an evening together with friends and also the legendary "Green Fairy". |
| Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first. The following rings are all real closed rings:
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| * [[real closed field]]s. These are exactly the real closed rings that are fields.
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| * the ring of all [[Tychonoff_space#Real-valued continuous functions|real valued continuous functions]] on a [[completely regular space]] ''X''. Also, the ring of all bounded real valued continuous functions on ''X'' is real closed.
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| * convex subrings of real closed fields. These are precisely those real closed rings which are also [[valuation ring]]s and were initially studied by Cherlin and Dickmann (they used the term 'real closed ring' for what is now called 'real closed valuation ring').
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| * the ring ''A'' of all continuous [[semialgebraic set|semi-algebraic function]]s on a semi-algebraic set of a real closed field (with values in that field). Also, the subring of all bounded (in any sense) functions in ''A'' is real closed.
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| * (generalizing the previous example) the ring of all (bounded) continuous definable functions on a [[definable set]] ''S'' of an arbitrary first-order [[expansion of a first-order structure|expansion]] ''M'' of a real closed field (with values in ''M''). Also, the ring of all (bounded) definable functions <math>S\to M</math> is real closed.
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| * Real closed rings are precisely the rings of [[global section]]s of affine real closed spaces (a generalization of [[semialgebraic space]]s) and in this context they were invented by Niels Schwartz in the early 1980s.
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| ==Definition==
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| A real closed ring is a reduced, commutative unital ring ''A'' which has the following properties:
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| #The set of squares of ''A'' is the set of nonnegative elements of a partial order ≤ on ''A'' and ''(A,≤)'' is an [[f-ring]].
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| #Convexity condition: For all a,b from ''A'', if 0≤a≤b then b|a<sup>2</sup>.
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| #For every [[prime ideal]] ''p'' of ''A'', the [[residue class ring]] ''A/p'' is [[integrally closed]] and its [[field of fractions]] is a real closed field.
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| The link to the definition at the beginning of this article is given in the section on algebraic properties below. | |
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| ==The real closure of a commutative ring==
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| Every commutative unital ring ''R'' has a so-called '''real closure''' rcl(''R'') and this is unique up to a unique ring
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| homomorphism over ''R''. This means that rcl(''R'') is a real closed ring and there is a (not necessarily injective) ring homomorphism
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| <math>r:R\to rcl(R)</math> such that for every ring homomorphism <math>f:R\to A</math> to some other real closed ring ''A'', there is a unique ring homomorphism <math>g:rcl(R)\to A</math> with <math>f=g\circ r</math>.
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| For example the real closure of the polynomial ring <math>\mathbb{R}[T_1,...,T_n]</math>
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| is the ring of continuous semi-algbebraic functions <math>\mathbb{R}^n\to \mathbb{R}</math>.
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| Note that an arbitrary ring ''R'' is semi-real (i.e. -1 is not a sum of squares in ''R'')
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| if and only if the real closure of ''R'' is not the null ring.
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| Note also that the real closure of an [[ordered field]] is in general '''not''' the real closure of the underlying field. For example, the real closure of the '''ordered''' subfield <math>\mathbb{Q}(\sqrt 2)</math>
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| of <math>\mathbb{R}</math> is the field <math>\mathbb{R}_{alg}</math> of real algebraic numbers,
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| whereas the real closure of the field <math>\mathbb{Q}(\sqrt 2)</math> is the ring
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| <math>\mathbb{R}_{alg}\times \mathbb{R}_{alg}</math> (corresponding to the two orders of <math>\mathbb{Q}(\sqrt 2)</math>). More generally the real closure of a field ''F''
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| is a certain subdirect product of the real closures of the ordered fields ''(F,P)'', where ''P'' runs through the orderings of ''F''. | |
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| ==Algebraic properties==
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| * The [[category theory|category]] ''RCR'' of real closed rings which has real closed rings as objects and ring homomorphisms as maps has the following properties:
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| #Arbitrary products, direct limits and inverse limits (in the category of commutative unital rings) of real closed rings are again real closed. The [[Pushout (category theory)|fibre sum]] of two real closed rings ''B,C'' over some real closed ring ''A'' exists in ''RCR'' and is the real closure of the [[tensor product]] of ''B'' and ''C'' over ''A''.
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| #''RCR'' has arbitrary limits and co-limits.
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| #''RCR'' is a [[Variety (universal algebra)|variety]] in the sense of [[universal algebra]] (but not a subvariety of commutative rings).
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| * For a real closed ring ''A'', the natural homomorphism of ''A'' to the product of all its [[residue field]]s is an isomorphism onto a subring of this product that is closed under continuous [[Semialgebraic set|semi-algebraic]] functions defined over the integers. Conversely, every subring of a product of real closed fields with this property is real closed.
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| * If ''I'' is a [[radical ideal]] of a real closed ring ''A'', then also the [[residue class ring]] ''A''/''I'' is real closed. If ''I'' and ''J'' are radical ideals of a real closed ring then the sum ''I'' + ''J'' is again a radical ideal.
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| * All classical [[Localization of a ring|localization]]s ''S''<sup>−1</sup>''A'' of a real closed ring ''A'' are real closed. The epimorphic hull and the complete ring of quotients of a real closed ring are again real closed.
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| * The (real) holomorphy ring ''H''(''A'') of a real closed ring ''A'' is again real closed. By definition, ''H''(''A'') consists of all elements ''f'' in ''A'' with the property ''−N'' ≤ ''f'' ≤ ''N'' for some natural number ''N''. Applied to the examples above, this means that the rings of bounded (semi-algberaic/definable) continuous functions are all real closed.
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| * The support map from the [[real spectrum]] of a real closed ring to its [[Spectrum of a ring|Zariski spectrum]], which sends an ordering ''P'' to its support <math>P\cap -P</math> is an [[homeomorphism]]. In particular, the Zariski spectrum of every real closed ring ''A'' is a root system (in the sense of [[graph theory]]) and therefore ''A'' is also a Gel'fand ring (i.e. every [[prime ideal]] of ''A'' is contained in a unique maximal ideal of ''A''). The comparison of the Zariski spectrum of ''A'' with the Zariski spectrum of ''H(A)'' leads to a homeomorphism between the maximal spectra of these rings, generalizing the Gel'fand-Kolmogorov theorem for rings of real valued continuous functions.
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| * The natural map ''r'' from an arbitrary ring ''R'' to its real closure rcl(''R'') as explained above, induces a homeomorphism from the real spectrum of rcl(''R'') to the real spectrum of ''R''.
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| * Summarising and significantly strengthening the previous two properties, the following is true: The natural map ''r'' from an arbitrary ring ''R'' to its real closure rcl(''R'') induces an identification of the [[affine scheme]] of rcl(''R'') with the affine real closed space of ''R''.
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| ==Model theoretic properties==
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| The class of real closed rings is [[first-order]] [[axiom]]atizable and [[Decidability (logic)|undecidable]]. The class of all real closed valuation rings is [[Decidability (logic)|decidable]] (by Cherlin-Dickmann) and the class of all real closed fields is decidable (by Tarski). After naming a definable radical relation, real closed rings have a [[Model complete theory|model companion]], namely [[von Neumann regular]] real closed rings.
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| ==Comparison with characterizations of real closed fields==
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| There are many different characterizations of [[real closed field|real closed '''fields''']]. For example
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| in terms of maximality (with respect to algebraic extensions): a real closed field is a maximally orderable field; or, a real closed field (together with its unique ordering) is a maximally ordered field. Another characterization says that the intermediate value theorem holds for all polynomials in one variable over the (ordered) field. In the case of commutative rings, all these properties can be (and are) analyzed in the literature. They all lead to different classes of rings which are unfortunately also called 'real closed' (because a certain characterization of real closed fields has been extended to rings). '''None''' of them lead to the class of real closed rings and none of them allow a satisfactory notion of a closure operation. A central point in the definition of real closed rings is the globalisation of the notion of a real closed field to rings when these rings are represented as rings of functions on some space (typically, the real spectrum of the ring).
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| == References ==
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| <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using<ref></ref> tags which will then appear here automatically --> | |
| {{Reflist}}
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| * Cherlin, Gregory. Rings of continuous functions: decision problems Model theory of algebra and arithmetic (Proc. Conf., Karpacz, 1979), pp. 44–91, Lecture Notes in Math., 834, Springer, Berlin, 1980.
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| * Cherlin, Gregory(1-RTG2); Dickmann, Max A. Real closed rings. II. Model theory. Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231.
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| * A. Prestel, N. Schwartz. Model theory of real closed rings. Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 261–290, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002.
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| * Schwartz, Niels. The basic theory of real closed spaces. Memoirs of the American Mathematical Society 1989 (ISBN 0821824600 )
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| * Schwartz, Niels; Madden, James J. Semi-algebraic function rings and reflectors of partially ordered rings. Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin, 1999
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| * Schwartz, Niels. Real closed rings. Algebra and order (Luminy-Marseille, 1984), 175–194, Res. Exp. Math., 14, Heldermann, Berlin, 1986
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| * Schwartz, Niels. Rings of continuous functions as real closed rings. Ordered algebraic structures (Curaçao, 1995), 277–313, Kluwer Acad. Publ., Dordrecht, 1997.
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| * Tressl, Marcus. Super real closed rings. Fundamenta Mathematicae 194 (2007), no. 2, 121–177.
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| <!--- Categories --->
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| [[Category:Ring theory]]
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| [[Category:Real algebraic geometry]]
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| [[Category:Ordered algebraic structures]]
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| [[Category:Model theory]]
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| [[Category:Real closed field]]
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Buying absinthe online is significantly less easy as walking right into a liquor store and getting a bottle of absinthe. Although a little difficult, buying absinthe on the internet is still possible and you've got to have few precautions. If you have any kind of inquiries pertaining to where and how to use get The facts, you could call us at our own website. US citizens face difficulty when they have got to buy absinthe online, as thujone that's an essential active ingredient in absinthe is forbidden in the United States. However, it's been demonstrated that thujone doesn't cause hallucinations similar to cannabis as earlier considered.
youtube.comProduction and sale of absinthe in the United States is strictly prohibited; however, importing absinthe that does not consist of any traces of thujone is permitted. For purchasing any type of alcohol you should be 21 years or older and anyone found violating this law is punished. There are many who are thinking about buying absinthe in the United States. So what is it that pulls lovers of fine liquor to absinthe?
Well, absinthe is regarded as the famous and controversial liquor made from herbs like wormwood (Artemisia absinthium), green anise, fennel seed, angelica root, star anise hyssop, melissa, and peppermint leaves. The herb wormwood is the most important ingredient and it consists of thujone that's answerable for absinthe unique effects. Absinthe liquor is bitter to taste thus it's always taken with sugar and ice-cold water. Absinthe drink is geared up using a traditional ritual.
In the traditional French ritual a special absinthe glass, absinthe spoon, sugar cube, and ice-cold water is needed. An ounce or two of absinthe is put in the glass and an absinthe spoon is kept over the rim of the glass. The absinthe spoon has perforations in its basin. A sugar cube click this link is kept on the spoon and ice-cold water is poured over the cube from an absinthe fountain, as the water melts the sugar cube it falls through the perforations into the glass. The clear green absinthe becomes turbid or milky white for the reason that essential oils from your herbs get precipitated. This is whats called louching of absinthe. Some more ice-cold water is added and the drink is prepared. Absinthe was extremely popular in Europe in the 18th and nineteenth century and was adoringly known as "The Green Fairy" or "Green Muse". However, due to specific misconceptions and rumors Visit us at absinthe was prohibited in Europe and North America during the 20th century.
Within the light of new evidence introduced by scientists in the year 1999 most European countries removed the ban on absinthe and it is now accessible to people. In the United States it is still prohibited; however, US citizens can buy absinthe online from a variety of online stores. Absinthe is a powerful alcoholic drink and the alcohol content is as high as 60%.
Since buying real absinthe is against the law in the US many replicas of absinthe can be bought online. You ought to be cautious with such fraudulent websites which promise to ship you the real thing. You can buy absinthe online from a number of the most respected distillers in Europe. The absinthe shipped will be minus thujone and perfectly legal. Ordering absinthe from French distilleries is also a wise decision and the product you get does not consist of thujone. Even so, when ordering absinthe online you need to be cautious as some websites may be illegal.
Possibly the most suitable option intended for US citizens is to buy absinthe kits from non-US producers and ready your own absinthe in your own home. The kit includes all of the essential herbs, recommendations on preparation, rituals along with other accessories just like spoon, straining cloth, etc you can prepare about 2 liters of absinthe liquor from one kit that will keep a party of eight going until dawn. Serving absinthe at a party may also enable you to get the title "King of good times". One place where you can be assured of high quality herbs and genuine absinthe essence is absinthekit.com
You should order your kit and spend an evening together with friends and also the legendary "Green Fairy".