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| In [[commutative algebra]], an '''integrally closed domain''' ''A'' is an [[integral domain]] whose [[integral closure]] in its field of fractions is ''A'' itself. Many well-studied domains are integrally closed: [[Field (mathematics)|Field]]s, the ring of integers '''Z''', [[unique factorization domain]]s and regular local rings are all integrally closed.
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. You can either install Word - Press yourself or use free services offered on the web today. The effect is to promote older posts by moving them back onto the front page and into the rss feed. Hosted by Your Domain on Another Web Host - In this model, you first purchase multiple-domain webhosting, and then you can build free Wordpress websites on your own domains, taking advantage of the full power of Wordpress. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. This plugin is a must have for anyone who is serious about using Word - Press. You can up your site's rank with the search engines by simply taking a bit of time with your site. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>Usually, Wordpress owners selling the ad space on monthly basis and this means a residual income source. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. This platform can be customizedaccording to the requirements of the business. The first thing you need to do is to choose the right web hosting plan. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. * Robust CRM to control and connect with your subscribers. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If you have any concerns regarding where and how you can use [http://urlx.at/backupplugin919874 wordpress dropbox backup], you could contact us at our web site. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. Visit our website to learn more about how you can benefit. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. Customers within a few seconds after visiting a site form their opinion about the site. |
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| To give a non-example,<ref>Taken from Matsumura</ref> let <math>A = k[t^2, t^3] \subset B = k[t]</math> (''k'' a field). ''A'' and ''B'' have the same field of fractions, and ''B'' is the integral closure of ''A'' (since ''B'' is a UFD.) In other words, ''A'' is not integrally closed. This is related to the fact that the plane curve <math>Y^2 = X^3</math> has a singularity at the origin.
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| Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a finite extension of ''K''. Then ''x'' in ''L'' is integral over ''A'' if and only if its minimal polynomial over ''K'' has coefficients in ''A''.<ref>Matsumura, Theorem 9.2</ref> This implies in particular that an integral element over an integrally closed domain ''A'' has a minimal polynomial over ''A''. This is stronger than the statement that any integral element satisfies some monic polynomial. In fact, the statement is false without "integrally closed" (consider <math>A = \mathbb{Z}[\sqrt{5}].</math>)
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| Integrally closed domains also play a role in the hypothesis of the [[Going-down theorem]]. The theorem states that if ''A''⊆''B'' is an [[integral extension]] of domains and ''A'' is an integrally closed domain, then the [[going up and going down|going-down property]] holds for the extension ''A''⊆''B''.
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| Note that integrally closed domain appear in the following chain of [[subclass (set theory)|class inclusions]]:
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| : '''[[Commutative ring]]s''' ⊃ '''[[integral domain]]s''' ⊃ '''integrally closed domains''' ⊃ '''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''
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| == Examples ==
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| The following are integrally closed domains.
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| *Any principal ideal domain (in particular, any field).
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| *Any [[unique factorization domain]] (in particular, any polynomial ring over a unique factorization domain.)
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| *Any [[GCD domain]] (in particular, any [[Bézout domain]] or [[valuation domain]]).
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| *Any [[Dedekind domain]].
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| *Any [[symmetric algebra]] over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field).
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| == Noetherian integrally closed domain ==
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| For a noetherian local domain ''A'' of dimension one, the following are equivalent.
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| *''A'' is integrally closed.
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| *The maximal ideal of ''A'' is principal.
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| *''A'' is a [[discrete valuation ring]] (equivalently ''A'' is Dedekind.)
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| *''A'' is a regular local ring.
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| Let ''A'' be a noetherian integral domain. Then ''A'' is integrally closed if and only if (i) ''A'' is the intersection of all localizations <math>A_\mathfrak{p}</math> over prime ideals <math>\mathfrak{p}</math> of height 1 and (ii) the localization <math>A_\mathfrak{p}</math> at a prime ideal <math>\mathfrak{p}</math> of height 1 is a discrete valuation ring.
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| A noetherian ring is a [[Krull domain]] if and only if it is an integrally closed domain.
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| In the non-noetherian setting, one has the following: an integral domain is integrally closed if and only if it is the intersection of all [[valuation ring]]s containing it.
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| == Normal rings ==
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| {{See also|normal variety}}
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| Authors including [[Jean-Pierre Serre|Serre]], [[Alexander Grothendieck|Grothendieck]], and Matsumura define a '''normal ring''' to be a ring whose localizations at prime ideals are integrally closed domains. Such a ring is necessarily a [[reduced ring]],<ref>If all localizations at maximal ideals of a commutative ring ''R'' are reduced rings (e.g. domains), then ''R'' is reduced. ''Proof'': Suppose ''x'' is nonzero in ''R'' and ''x''<sup>2</sup>=0. The [[annihilator]] ann(''x'') is contained in some maximal ideal <math>\mathfrak{m}</math>. Now, the image of ''x'' is nonzero in the localization of ''R'' at <math>\mathfrak{m}</math> since <math>x = 0</math> at <math>\mathfrak{m}</math> means <math>xs = 0</math> for some <math>s \not\in \mathfrak{m}</math> but then <math>s</math> is in the annihilator of ''x'', contradiction. This shows that ''R'' localized at <math>\mathfrak{m}</math> is not reduced.</ref> and this is sometimes included in the definition. In general, if ''A'' is a [[Noetherian ring|Noetherian]] ring whose localizations at maximal ideals are all domains, then ''A'' is a finite product of domains.<ref>Kaplansky, Theorem 168, pg 119.</ref> In particular if ''A'' is a Noetherian, normal ring, then the domains in the product are integrally closed domains.<ref>Matsumura 1989, p. 64</ref> Conversely, any finite product of integrally closed domains is normal. In particular, if <math>\operatorname{Spec}(A)</math> is noetherian, normal and connected, then ''A'' is an integrally closed domain. (cf. [[smooth variety]])
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| Let ''A'' be a noetherian ring. Then ''A'' is normal if and only if it satisfies the following: for any prime ideal <math>\mathfrak{p}</math>,
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| *(i) If <math>\mathfrak{p}</math> has height <math>\le 1</math>, then <math>A_\mathfrak{p}</math> is [[regular local ring|regular]] (i.e., <math>A_\mathfrak{p}</math> is a [[discrete valuation ring]].)
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| *(ii) If <math>\mathfrak{p}</math> has height <math>\ge 2</math>, then <math>A_\mathfrak{p}</math> has depth <math>\ge 2</math>.<ref>Matsumura, Commutative algebra, pg. 125. For a domain, the theorem is due to Krull (1931). The general case is due to Serre.</ref>
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| Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of [[associated prime]]s <math>Ass(A)</math> has no [[embedded prime]]s, and, when (i) is the case, (ii) means that <math>Ass(A/fA)</math> has no embedded prime for any nonzero zero-divisor ''f''. In particular, a [[Cohen-Macaulay ring]] satisfies (ii). Geometrically, we have the following: if ''X'' is a [[local complete intersection]] in a nonsingular variety;<ref>over an algebraically closed field</ref> e.g., ''X'' itself is nonsingular, then ''X'' is Cohen-Macaulay; i.e., the stalks <math>\mathcal{O}_p</math> of the structure sheaf are Cohen-Macaulay for all prime ideals p. Then we can say: ''X'' is [[normal scheme|normal]] (i.e., the stalks of its structure sheaf are all normal) if and only if it is regular in codimension ''1''.
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| == Completely integrally closed domains ==
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| Let ''A'' be a domain and ''K'' its field of fractions. ''x'' in ''K'' is said to be '''almost integral over ''A'' ''' if there is a <math>d \ne 0</math> such that <math>d x^n \in A</math> for all <math>n \ge 0</math>. Then ''A'' is said to be '''completely integrally closed''' if every almost integral element of ''K'' is contained in ''A''. A completely integrally closed domain is integrally closed. Conversely, a noetherian integrally closed domain is completely integrally closed.
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| Assume ''A'' is completely integrally closed. Then the formal power series ring <math>A[[X]]</math> is completely integrally closed.<ref>An exercise in Matsumura.</ref> This is significant since the analog is false for an integrally closed domain: let ''R'' be a valuation domain of height at least 2 (which is integrally closed.) Then <math>R[[X]]</math> is not integrally closed.<ref>Matsumura, Exercise 10.4</ref> Let ''L'' be a field extension of ''K''. Then the integral closure of ''A'' in ''L'' is completely integrally closed.<ref>An exercise in Bourbaki.</ref>
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| == "Integrally closed" under constructions ==
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| The following conditions are equivalent for an integral domain ''A'':
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| # ''A'' is integrally closed;
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| # ''A''<sub>''p''</sub> (the localization of ''A'' with respect to ''p'') is integrally closed for every [[prime ideal]] ''p'';
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| # ''A''<sub>''m''</sub> is integrally closed for every [[maximal ideal]] ''m''.
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| 1 → 2 results immediately from the preservation of integral closure under localization; 2 → 3 is trivial; 3 → 1 results from the preservation of integral closure under localization, the [[Localization of a module#Flatness|exactness of localization]], and the property that an ''A''-module ''M'' is zero if and only if its localization with respect to every maximal ideal is zero.
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| In contrast, the "integrally closed" does not pass over quotient, for '''Z'''[t]/(t<sup>2</sup>+4) is not integrally closed.
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| The localization of a completely integrally closed need not be completely integrally closed.<ref>An exercise in Bourbaki.</ref>
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| A direct limit of integrally closed domains is an integrally closed domain.
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| == Modules over an integrally closed domain ==
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| {{expand section|date=February 2013}}
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| == See also ==
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| *[[Unibranch local ring]]
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| == References ==
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| {{reflist}}
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| *Bourbaki, Commutative algebra.
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| * {{cite book | last = Kaplansky | first = Irving | title = Commutative Rings
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| | series = Lectures in Mathematics |date=September 1974
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| | publisher = [[University of Chicago Press]] | isbn = 0-226-42454-5 }}
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| *Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6.
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| *Matsumura, Hideyuki (1970) ''Commutative algebra'' ISBN 0-8053-7026-9.
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| [[Category:Commutative algebra]]
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