|
|
| Line 1: |
Line 1: |
| 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.
| | by Nas, is very fitting and the film agrees with it. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have. <br><br>Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. If you enjoyed this article and you would certainly such as to receive even more facts concerning [http://by.ix-cafe.com/wordpressdropboxbackup189314 wordpress dropbox backup] kindly browse through our own web-site. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>But before choosing any one of these, let's compare between the two. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>Numerous bloggers are utilizing Word - Press and with good reason. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Website security has become a major concern among individuals all over the world. Automated deal feed integration option to populate your blog with relevant deals. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Thus, Word - Press is a good alternative if you are looking for free blogging software. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins. |
| | |
| 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.
| |
| | |
| 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>)
| |
| | |
| 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''.
| |
| | |
| Note that integrally closed domain appear in the following chain of [[subclass (set theory)|class inclusions]]:
| |
| : '''[[Commutative ring]]s''' ⊃ '''[[integral domain]]s''' ⊃ '''integrally closed domains''' ⊃ '''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|field]]s'''
| |
| | |
| == Examples ==
| |
| The following are integrally closed domains.
| |
| *Any principal ideal domain (in particular, any field).
| |
| *Any [[unique factorization domain]] (in particular, any polynomial ring over a unique factorization domain.)
| |
| *Any [[GCD domain]] (in particular, any [[Bézout domain]] or [[valuation domain]]).
| |
| *Any [[Dedekind domain]].
| |
| *Any [[symmetric algebra]] over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field).
| |
| | |
| == Noetherian integrally closed domain ==
| |
| | |
| For a noetherian local domain ''A'' of dimension one, the following are equivalent.
| |
| *''A'' is integrally closed.
| |
| *The maximal ideal of ''A'' is principal.
| |
| *''A'' is a [[discrete valuation ring]] (equivalently ''A'' is Dedekind.)
| |
| *''A'' is a regular local ring.
| |
| | |
| 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.
| |
| | |
| A noetherian ring is a [[Krull domain]] if and only if it is an integrally closed domain.
| |
| | |
| 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.
| |
| | |
| == Normal rings ==
| |
| {{See also|normal variety}}
| |
| 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]])
| |
| | |
| 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>,
| |
| *(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]].)
| |
| *(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>
| |
| | |
| 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''.
| |
| | |
| == Completely integrally closed domains ==
| |
| 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.
| |
| | |
| 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>
| |
| | |
| == "Integrally closed" under constructions ==
| |
| The following conditions are equivalent for an integral domain ''A'':
| |
| # ''A'' is integrally closed;
| |
| # ''A''<sub>''p''</sub> (the localization of ''A'' with respect to ''p'') is integrally closed for every [[prime ideal]] ''p'';
| |
| # ''A''<sub>''m''</sub> is integrally closed for every [[maximal ideal]] ''m''.
| |
| | |
| 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.
| |
| | |
| In contrast, the "integrally closed" does not pass over quotient, for '''Z'''[t]/(t<sup>2</sup>+4) is not integrally closed. | |
| | |
| The localization of a completely integrally closed need not be completely integrally closed.<ref>An exercise in Bourbaki.</ref>
| |
| | |
| A direct limit of integrally closed domains is an integrally closed domain.
| |
| | |
| == Modules over an integrally closed domain ==
| |
| {{expand section|date=February 2013}}
| |
| | |
| == See also ==
| |
| *[[Unibranch local ring]]
| |
| | |
| == References ==
| |
| {{reflist}}
| |
| *Bourbaki, Commutative algebra.
| |
| * {{cite book | last = Kaplansky | first = Irving | title = Commutative Rings
| |
| | series = Lectures in Mathematics |date=September 1974
| |
| | publisher = [[University of Chicago Press]] | isbn = 0-226-42454-5 }}
| |
| *Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6.
| |
| *Matsumura, Hideyuki (1970) ''Commutative algebra'' ISBN 0-8053-7026-9.
| |
| | |
| [[Category:Commutative algebra]]
| |
by Nas, is very fitting and the film agrees with it. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Transforming your designs to Word - Press blogs is not that easy because of the simplified way in creating your very own themes. Also our developers are well convergent with the latest technologies and bitty-gritty of wordpress website design and promises to deliver you the best solution that you can ever have.
Generally, for my private income-making market websites, I will thoroughly research and discover the leading 10 most worthwhile niches to venture into. You will have to invest some money into tuning up your own blog but, if done wisely, your investment will pay off in the long run. It sorts the results of a search according to category, tags and comments. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. If you enjoyed this article and you would certainly such as to receive even more facts concerning wordpress dropbox backup kindly browse through our own web-site. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites.
But before choosing any one of these, let's compare between the two. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory.
Numerous bloggers are utilizing Word - Press and with good reason. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. Word - Press offers constant updated services and products, that too, absolutely free of cost.
Website security has become a major concern among individuals all over the world. Automated deal feed integration option to populate your blog with relevant deals. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. Thus, Word - Press is a good alternative if you are looking for free blogging software. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.