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| In [[mathematics]], in the subfield of [[ring theory]], a ring ''R'' is a '''polynomial identity ring''' if there is, for some ''N'' > 0, an element ''P'' other than 0 of the [[free algebra]], Z<''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>>, over the [[ring of integers]] in ''N'' variables ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub> such that for all ''N''-[[tuple]]s ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''N''</sub> taken from ''R'' it happens that
| | Katie Hopkins has insisted that �peacocking on social media� will not help save the kidnapped Nigerian school girls<br><br> |
| | The relentless motor mouth wrote about [http://tinyurl.com/kecvhhb http://tinyurl.com/kecvhhb] the harrowing issue in her latest Sun column, hitting out at celebrities who [http://tinyurl.com/kecvhhb cheap ugg boots] have been uploading photos of themselves with posters reading #BringBackOurGirls<br> |
| | �Listen fools,� she said. �You can hold up a sign with a hashtag until your arms drop off, but these girls are not coming back alive unless someone goes in and gets them.<br> |
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| :<math>P(r_1, r_2, \ldots, r_N) = 0.\ </math>
| | High-profile figures to take part in the selfie campaign so far have included US First Lady Michelle Obama, prime minister David Cameron and supermodel Cara Deleving<br>. |
| | Prime Minister David Cameron joins the campaign to to save the kidnapped schoolgir<br> |
| | But Hopkins has argued that �Obamageddon� would have kicked in to rescue the girls, had they been from Amer<br>a. |
| | �If 276 high school girls had been kidnapped from the US, Obamageddon would have been unleashed,� she wrote. �Instead, Mrs Obama has tried asking nicely, made big puppy dog eyes at the ca<br><br>. |
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| Strictly the ''X''<sub>''i''</sub> here are "non-commuting indeterminates", and so "polynomial identity" is a slight [[abuse of notation|abuse of language]], since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation '''PI-ring''' is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of '''PI-algebra'''.
| | �David Cameron wasn�t going to be outdone by Mrs Obama and got himself a sign with the same hashtag. Dave knows it is important to be seen to be doing something and, apparently, holding a sign is very much in vogue.� |
| | Hopkins� article has been met with positive responses online - a stark contrast to the usual reception reserved for her often-controversial and always outspoken op<br>ions. |
| | Don't normally agree with @KTHopkins but she's spot on this morning #stopholdingsignsandrescuet<br><br>rls |
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| If the degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called '''monic''' if at least one of its terms of highest degree has coefficient equal to 1.
| | - Sonja webster (@webster_sonja) May 16, 2014She's right RT @kthopkins You can hold up a sign until your arms drop off, those girls aren't coming back unless someone goes in and <br>ts em |
| | - Nick Southall (@Nick_Southall) May 16, 2014i don't favour a lot of her opinions but this [http://tinyurl.com/kecvhhb http://tinyurl.com/kecvhhb] time i'm agreeing with @KThopkins <br>bsite |
| | - RobK (@6soundmint6) May [http://tinyurl.com/kecvhhb cheap ugg boots] 1<br> 2014 |
| | The former Apprentice star emphasised the need for �giants of men with daggers on their berets and courage in their hearts to teach Mad Dog Boko Bill a <br><br>on�. |
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| Every commutative ring is a PI-ring, satisfying the polynomial identity ''XY'' - ''YX'' = 0. Therefore PI-rings are usually taken as ''close generalizations of commutative rings''. If the ring has [[Characteristic (algebra)|characteristic]] ''p'' different from zero then it satisfies the polynomial identity ''pX'' = 0. To exclude such examples, sometimes it is defined that PI-rings must satisfy a monic polynomial identity.<ref>J.C. McConnell, J.C. Robson, ''Noncommutative Noetherian Rings, Graduate studies in Mathematics, Vol 30''</ref>
| | Local residents have started taking matters into their own hands and [http://tinyurl.com/kecvhhb discount ugg boots] forming vigilante groups, after feeling that the Nigerian military was not doing enough to stem attacks by the Boko [http://tinyurl.com/kecvhhb ugg boots sale] Haram Islamic mi<br>tants. |
| | | Hopkins, meanwhile, spoke at the [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Cambridge+Union&Submit=Go Cambridge Union] earlier this week, where she told students she didn't "really like fat people" and "wouldn't like to be ginger in the dark". |
| ==Examples==
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| * For example if ''R'' is a [[commutative ring]] it is a PI-ring: this is true with
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| ::<math>P(X_1,X_2)=X_1X_2-X_2X_1=0~</math>
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| * A major role is played in the theory by the '''standard identity''' ''s''<sub>''N''</sub>, of length ''N'', which generalises the example given for commutative rings (''N'' = 2). It derives from the [[Leibniz formula for determinants]]
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| ::<math>\det(A) = \sum_{\sigma \in S_N} \sgn(\sigma) \prod_{i = 1}^N a_{i,\sigma(i)}</math>
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| :by replacing each product in the summand by the product of the ''X''<sub>i</sub> in the order given by the permutation σ. In other words each of the ''N''! orders is summed, and the coefficient is 1 or −1 according to the [[signature of a permutation|signature]].
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| ::<math>s_N(X_1,\ldots,X_N) = \sum_{\sigma \in S_N} \sgn(\sigma) X_{\sigma(1)}\dotsm X_{\sigma(N)}=0~</math>
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| :The ''k''×''k'' [[matrix ring]] over any commutative ring satisfies a standard identity; the [[Amitsur–Levitzki theorem]] states that it satisfies ''s''<sub>2''k''</sub>. The degree of this identity is optimal since the matrix ring can not satisfy any monic polynomial of degree less than 2''k''.
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| * Given a field ''k'' of characteristic zero, take ''R'' to be the [[exterior algebra]] over a [[countably infinite]]-dimensional [[vector space]] with basis ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>3</sub>, ... Then ''R'' is generated by the elements of this basis and
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| ::''e''<sub>''i''</sub>''e''<sub>''j''</sub> = −''e''<sub>''j''</sub>''e''<sub>''i''</sub>. | |
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| :This ring does not satisfy ''s''<sub>''N''</sub> for any ''N'' and therefore can not be embedded in any matrix ring. In fact ''s''<sub>''N''</sub>(''e''<sub>1</sub>,''e''<sub>2</sub>,...,''e''<sub>''N''</sub>) = ''N''!''e''<sub>1</sub>''e''<sub>2</sub>...''e''<sub>''N''</sub> ≠ 0. On the other hand it is a PI-ring since it satisfies [[''x'', ''y''], ''z''] := ''xyz'' − ''xzz'' − ''zxy'' + ''zyz'' = 0. It is enough to check this for monomials in the ''e'''s. Now, a monomial of even degree commutes with every element. Therefore if either ''x'' or ''y'' is a monomial of even degree [''x'', ''y''] := ''xy'' − ''yx'' = 0. If both are of odd degree then [''x'', ''y''] = ''xy'' − ''yx'' = 2''xy'' has even degree and therefore commutes with ''z'', i.e. [[''x'', ''y''], ''z''] = 0.
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| ==Properties== | |
| * Any [[subring]] or [[homomorphism|homomorphic image]] of a PI-ring is a PI-ring.
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| * A finite [[direct product]] of PI-rings is a PI-ring.
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| * A direct product of PI-rings, satisfying the same identity, is a PI-ring.
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| * It can always be assumed that the identity that the PI-ring satisfies is [[multilinear]].
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| * If a ring is finitely generated by ''n'' elements as a [[module (mathematics)|module]] over its [[Center (algebra)|center]] then it satisfies every alternating multilinear polynomial of degree larger than ''n''. In particular it satisfies ''s''<sub>''N''</sub> for ''N'' > ''n'' and therefore it is a PI-ring.
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| * If ''R'' and ''S'' are PI-rings then their [[tensor product]] over the integers, <math>R\otimes_\mathbb{Z}S</math>, is also a PI-ring.
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| * If ''R'' is a PI-ring, then so is the ring of ''n''×''n''-matrices with coefficients in ''R''.
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| ==PI-rings as generalizations of commutative rings==
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| Among noncommutative rings, PI-rings satisfy the [[Köthe conjecture]]. [[Finitely generated algebra|Affine]] PI-algebras over a [[field (mathematics)|field]] satisfy the [[Kurosh conjecture]], the [[Nullstellensatz]] and the [[catenary ring|catenary property]] for [[prime ideals]].
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| If ''R'' is a PI-ring and ''K'' is a subring of its center such that ''R'' is [[integral extension|integal over]] ''K'' then the [[Going up and going down|going up and going down properties]] for prime ideals of ''R'' and ''K'' are satisfied. Also the ''lying over'' property (If ''p'' is a prime ideal of ''K'' then there is a prime ideal ''P'' of ''R'' such that <math>p</math> is minimal over <math>P\cap K</math>) and the ''incomparability'' property (If ''P'' and ''Q'' are prime ideals of ''R'' and <math>P\subset Q</math> then <math>P\cap K\subset Q\cap K</math>) are satisfied.
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| <!--Add here results about integral extensions similar to commutative case.-->
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| ==The set of identities a PI-ring satisfies==
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| If ''F'' := Z<''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''N''</sub>> is the free algebra in ''N'' variables and ''R'' is a PI-ring satisfying the polynomial ''P'' in ''N'' variables, then ''P'' is in the [[Kernel (algebra)|kernel]] of any homomorphism
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| :<math>\tau</math>:F <math>\rightarrow</math> ''R''.
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| An ideal ''I'' of ''F'' is called '''T-ideal''' if <math>f(I)\subset I</math> for every [[endomorphism]] ''f'' of ''F''.
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| Given a PI-ring, ''R'', the set of all polynomial identities it satisfies is an [[Ideal (ring theory)|ideal]] but even more it is a T-ideal. Conversely, if ''I'' is a T-ideal of ''F'' then ''F''/''I'' is a PI-ring satisfying all identities in ''I''. It is assumed that ''I'' contains monic polynomials when PI-rings are required to satisfy monic polynomial identities.
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| ==References==
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| {{Reflist}}
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| *{{eom|id=P/p072640|title=PI-algebra|first=V.N.|last= Latyshev}}
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| *{{eom|id=a/a110570|title=Amitsur–Levitzki theorem |first=E.|last= Formanek}}
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| *[http://books.google.com/books?id=Li147JZ4T6AC Polynomial identities in ring theory], Louis Halle Rowen, Academic Press, 1980, ISBN 978-0-12-599850-5
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| *[http://books.google.com/books?id=x8gJw5bMw2oC Polynomial identity rings], Vesselin S. Drensky, Edward Formanek, Birkhäuser, 2004, ISBN 978-3-7643-7126-5
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| *[http://books.google.com/books?id=ZLW_Kz_zOP8C Polynomial identities and asymptotic methods], A. Giambruno, Mikhail Zaicev, AMS Bookstore, 2005, ISBN 978-0-8218-3829-7
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| *[http://books.google.com/books?id=80pw1QoLSQUC Computational aspects of polynomial identities], Alexei Kanel-Belov, Louis Halle Rowen, A K Peters Ltd., 2005, ISBN 978-1-56881-163-5
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| ==External links==
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| *{{PlanetMath|urlname=PolynomialIdentityAlgebra|title=Polynomial identity algebra}}
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| *{{PlanetMath|urlname=StandardIdentity|title=Standard Identity}}
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| *{{PlanetMath|urlname=TIdeal|title=T-ideal}}
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| {{DEFAULTSORT:Polynomial Identity Ring}}
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| [[Category:Ring theory]]
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| [[Category:Mathematical identities]]
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Katie Hopkins has insisted that �peacocking on social media� will not help save the kidnapped Nigerian school girls
The relentless motor mouth wrote about http://tinyurl.com/kecvhhb the harrowing issue in her latest Sun column, hitting out at celebrities who cheap ugg boots have been uploading photos of themselves with posters reading #BringBackOurGirls
�Listen fools,� she said. �You can hold up a sign with a hashtag until your arms drop off, but these girls are not coming back alive unless someone goes in and gets them.
High-profile figures to take part in the selfie campaign so far have included US First Lady Michelle Obama, prime minister David Cameron and supermodel Cara Deleving
.
Prime Minister David Cameron joins the campaign to to save the kidnapped schoolgir
But Hopkins has argued that �Obamageddon� would have kicked in to rescue the girls, had they been from Amer
a.
�If 276 high school girls had been kidnapped from the US, Obamageddon would have been unleashed,� she wrote. �Instead, Mrs Obama has tried asking nicely, made big puppy dog eyes at the ca
.
�David Cameron wasn�t going to be outdone by Mrs Obama and got himself a sign with the same hashtag. Dave knows it is important to be seen to be doing something and, apparently, holding a sign is very much in vogue.�
Hopkins� article has been met with positive responses online - a stark contrast to the usual reception reserved for her often-controversial and always outspoken op
ions.
Don't normally agree with @KTHopkins but she's spot on this morning #stopholdingsignsandrescuet
rls
- Sonja webster (@webster_sonja) May 16, 2014She's right RT @kthopkins You can hold up a sign until your arms drop off, those girls aren't coming back unless someone goes in and
ts em
- Nick Southall (@Nick_Southall) May 16, 2014i don't favour a lot of her opinions but this http://tinyurl.com/kecvhhb time i'm agreeing with @KThopkins
bsite
- RobK (@6soundmint6) May cheap ugg boots 1
2014
The former Apprentice star emphasised the need for �giants of men with daggers on their berets and courage in their hearts to teach Mad Dog Boko Bill a
on�.
Local residents have started taking matters into their own hands and discount ugg boots forming vigilante groups, after feeling that the Nigerian military was not doing enough to stem attacks by the Boko ugg boots sale Haram Islamic mi
tants.
Hopkins, meanwhile, spoke at the Cambridge Union earlier this week, where she told students she didn't "really like fat people" and "wouldn't like to be ginger in the dark".