|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]], a '''real closed ring''' is a [[commutative ring]] ''A'' that
| |
| is a subring of a product of [[real closed field]]s, which is closed under
| |
| continuous [[Semialgebraic set|semi-algebraic]] functions defined over the integers.
| |
|
| |
|
| == Examples of real closed rings ==
| |
| 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:
| |
| * [[real closed field]]s. These are exactly the real closed rings that are fields.
| |
| * 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.
| |
| * 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').
| |
| * 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.
| |
| * (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.
| |
| * 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.
| |
|
| |
|
| ==Definition==
| | Graham er en av de skuespillerne som startet ut med et odor (no pun meant) som Roller Lady in Boogie Nights, males hvis karriere synes å ha vansmektet i dårlige filmer siden da. Jeg var nysgjerrig på å vite hva slags et clearly show ville snare den talentfulle underachiever å hoppe fra storskjerm til compact.And ABC ble tilsynelatende satse gården [http://Photobucket.com/images/p%C3%A5+Emily på Emily] evne til å oppfylle vår Carrie Bradshaw jones: Det var utallige annonser på Television, i papirutgave og på reklameplakater. Med en mektig metning, Emily hadde premiere på en mandag kveld i begynnelsen av januar.<br><br>Verktøy: boremotor, en 3/8 borekrone (for boltene) og en one/2 borekrone (lage et hull for 12 mm metallrør som holder laser) for unexciting av tre. Wire cutter (diker), ideelt som kommer til en liten skarp spiss. Radio Shack selger nummer for å gjøre Laptop bord genser lodding jobbe litt enklere, prøv en Radio Shack og noen thirty gauge som går med det. 4. Har et budsjett. For å få mest ut av pengene dine, må du ha en idé om hvor mye du er villig til å bruke.<br><br>Tere er foreløpig bare én aktiv leieavtale, som alle oth ers har utløpt. Når det var tid for styret å diskutere godkjenning av kjøp av en politi har kjøretøyet for å erstatte den som ble totalt i en oktober hendelsen, politimester Invoice Sala lys oppmuntrende oered, Vi tapte bare én [kjøretøyet], fylket mistet tre. Te plussiden er at det var den eldste bilen i eet. Te Saken ble godkjent uten konkurranse.<br><br>If you adored this article and you simply would like to get more info pertaining to [http://tinyurl.com/nb4bgts billige uggs sko] kindly visit the web site. |
| A real closed ring is a reduced, commutative unital ring ''A'' which has the following properties:
| |
| #The set of squares of ''A'' is the set of nonnegative elements of a partial order ≤ on ''A'' and ''(A,≤)'' is an [[f-ring]].
| |
| #Convexity condition: For all a,b from ''A'', if 0≤a≤b then b|a<sup>2</sup>.
| |
| #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.
| |
| The link to the definition at the beginning of this article is given in the section on algebraic properties below.
| |
| | |
| ==The real closure of a commutative ring==
| |
| Every commutative unital ring ''R'' has a so-called '''real closure''' rcl(''R'') and this is unique up to a unique ring
| |
| homomorphism over ''R''. This means that rcl(''R'') is a real closed ring and there is a (not necessarily injective) ring homomorphism
| |
| <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>.
| |
| | |
| For example the real closure of the polynomial ring <math>\mathbb{R}[T_1,...,T_n]</math>
| |
| is the ring of continuous semi-algbebraic functions <math>\mathbb{R}^n\to \mathbb{R}</math>.
| |
| | |
| Note that an arbitrary ring ''R'' is semi-real (i.e. -1 is not a sum of squares in ''R'')
| |
| if and only if the real closure of ''R'' is not the null ring.
| |
| | |
| 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>
| |
| of <math>\mathbb{R}</math> is the field <math>\mathbb{R}_{alg}</math> of real algebraic numbers,
| |
| whereas the real closure of the field <math>\mathbb{Q}(\sqrt 2)</math> is the ring
| |
| <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''
| |
| is a certain subdirect product of the real closures of the ordered fields ''(F,P)'', where ''P'' runs through the orderings of ''F''.
| |
| | |
| ==Algebraic properties==
| |
| * 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:
| |
| #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''.
| |
| #''RCR'' has arbitrary limits and co-limits.
| |
| #''RCR'' is a [[Variety (universal algebra)|variety]] in the sense of [[universal algebra]] (but not a subvariety of commutative rings).
| |
| * 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.
| |
| * 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.
| |
| * 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.
| |
| * 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.
| |
| * 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.
| |
| * 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''.
| |
| * 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''.
| |
| | |
| ==Model theoretic properties==
| |
| | |
| 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.
| |
| | |
| ==Comparison with characterizations of real closed fields==
| |
| | |
| There are many different characterizations of [[real closed field|real closed '''fields''']]. For example
| |
| 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).
| |
| | |
| == References ==
| |
| <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using<ref></ref> tags which will then appear here automatically -->
| |
| {{Reflist}}
| |
| * 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.
| |
| * Cherlin, Gregory(1-RTG2); Dickmann, Max A. Real closed rings. II. Model theory. Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231.
| |
| * 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.
| |
| * Schwartz, Niels. The basic theory of real closed spaces. Memoirs of the American Mathematical Society 1989 (ISBN 0821824600 )
| |
| * Schwartz, Niels; Madden, James J. Semi-algebraic function rings and reflectors of partially ordered rings. Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin, 1999
| |
| * Schwartz, Niels. Real closed rings. Algebra and order (Luminy-Marseille, 1984), 175–194, Res. Exp. Math., 14, Heldermann, Berlin, 1986
| |
| * Schwartz, Niels. Rings of continuous functions as real closed rings. Ordered algebraic structures (Curaçao, 1995), 277–313, Kluwer Acad. Publ., Dordrecht, 1997.
| |
| * Tressl, Marcus. Super real closed rings. Fundamenta Mathematicae 194 (2007), no. 2, 121–177.
| |
| | |
| <!--- Categories --->
| |
| [[Category:Ring theory]]
| |
| [[Category:Real algebraic geometry]]
| |
| [[Category:Ordered algebraic structures]]
| |
| [[Category:Model theory]]
| |
| [[Category:Real closed field]]
| |
Graham er en av de skuespillerne som startet ut med et odor (no pun meant) som Roller Lady in Boogie Nights, males hvis karriere synes å ha vansmektet i dårlige filmer siden da. Jeg var nysgjerrig på å vite hva slags et clearly show ville snare den talentfulle underachiever å hoppe fra storskjerm til compact.And ABC ble tilsynelatende satse gården på Emily evne til å oppfylle vår Carrie Bradshaw jones: Det var utallige annonser på Television, i papirutgave og på reklameplakater. Med en mektig metning, Emily hadde premiere på en mandag kveld i begynnelsen av januar.
Verktøy: boremotor, en 3/8 borekrone (for boltene) og en one/2 borekrone (lage et hull for 12 mm metallrør som holder laser) for unexciting av tre. Wire cutter (diker), ideelt som kommer til en liten skarp spiss. Radio Shack selger nummer for å gjøre Laptop bord genser lodding jobbe litt enklere, prøv en Radio Shack og noen thirty gauge som går med det. 4. Har et budsjett. For å få mest ut av pengene dine, må du ha en idé om hvor mye du er villig til å bruke.
Tere er foreløpig bare én aktiv leieavtale, som alle oth ers har utløpt. Når det var tid for styret å diskutere godkjenning av kjøp av en politi har kjøretøyet for å erstatte den som ble totalt i en oktober hendelsen, politimester Invoice Sala lys oppmuntrende oered, Vi tapte bare én [kjøretøyet], fylket mistet tre. Te plussiden er at det var den eldste bilen i eet. Te Saken ble godkjent uten konkurranse.
If you adored this article and you simply would like to get more info pertaining to billige uggs sko kindly visit the web site.