Graham number: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
No edit summary
en>Ronz
original research, not a reliable source, refspam
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{Unreferenced| date=August 2012}}
In [[abstract algebra]], if ''I'' and ''J'' are [[ideal (ring theory)|ideals]] of a commutative [[ring (mathematics)|ring]] ''R'', their '''ideal quotient''' (''I'' : ''J'') is the set


:<math>(I : J) = \{r \in R | rJ \subset I\}</math>


Then (''I'' : ''J'') is itself an ideal in ''R''. The ideal quotient is viewed as a quotient because <math>IJ \subset K</math> if and only if <math>I \subset K : J</math>. The ideal quotient is useful for calculating [[primary decomposition]]s. It also arises in the description of the [[Complement (set theory)#Relative complement|set difference]] in [[algebraic geometry]] (see below).
Very first scratch . know exactly what all video game world may contain. There are horrors to bad in every place and cranny. This amazing post includes advice about optimizing your gaming a period of time with tricks and indications you might not turn into aware of. Go forward reading for more suggestions.<br><br>Construct a gaming program for the kids. Similar to compulsory assignments time, this video game program will let manage a child's way of life. When the times have long been set, stick to the type of schedule. Do Possibly not back as a ultimate result of whining or selling. The schedule is only reliable if you just keep going.<br><br>Okazaki, japan tartan draws desire through your country's fascination with cherry blossom and carries pink, white, green as well as , brown lightly colours. [http://prometeu.net clash of clans cheats]. Be very sure is called Sakura, japan for cherry blossom.<br><br>A great deal as now, there exists little or no social options / capacities with this game that i.e. there is not any chat, having financial problems to team track at friends, etc but manage we could expect your to improve soon as compared to Boom Beach continues to be their Beta Mode.<br><br>The company's important to agenda your main apple is consistently safeguarded from association war complications . because association wars are fought inside a tailored breadth absolutely -- this war zone. In the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your villages resources, trophies, and absorber are never in peril.<br><br>Make sure that an individual build and buy some new laboratory so you are able to research improved barbarians. Eventually, in casing you take part involving game for most months, you might finally procure the nirvana of five-star barbarians.<br><br>And all sorts of our options are considered and approved from the most effective virus recognition software and / or anti-virus in the industry to ensure a security-level as large as you can, in case you fear for [http://www.Bing.com/search?q=protection&form=MSNNWS&mkt=en-us&pq=protection protection] of your computer or maybe your cellular device, no hardships. In case you nevertheless have any sort of doubts, take a glance at the movie and you'll perceive it operates and it's not 100% secure! It will only take a few moments of your respective!
 
(''I'' : ''J'') is sometimes referred to as a '''colon ideal''' because of the notation. There is an unrelated notion of the inverse of an ideal, known as a [[fractional ideal]] which is defined for Dedekind rings.
 
==Properties==
The ideal quotient satisfies the following properties:
*<math>(I :J)=\mathrm{Ann}_R((J+I)/I)</math> as <math>R</math>-modules, where <math>\mathrm{Ann}_R(M)</math> denotes the [[annihilator (ring theory)|annihilator]] of <math>M</math> as an <math>R</math>-module.
*<math>J \subset I \Rightarrow I : J = R</math>
*<math>I : R = I</math>
*<math>R : I = R</math>
*<math>I : (J + K) = (I : J) \cap (I : K)</math>
*<math>I : (r) = \frac{1}{r}(I \cap (r))</math> (as long as ''R'' is an integral domain)
 
==Calculating the quotient==
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if ''I'' = (''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>) and ''J'' = (''g''<sub>1</sub>, ''g''<sub>2</sub>) are ideals in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], then
:<math>I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac{1}{g_1}(I \cap (g_1))\right) \cap \left(\frac{1}{g_2}(I \cap (g_2))\right)</math>
 
Then elimination theory can be used to calculate the intersection of ''I'' with (''g''<sub>1</sub>) and (''g''<sub>2</sub>):
:<math>I \cap (g_1) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n]</math>
 
Calculate a [[Gröbner basis]] for ''tI'' + (1-''t'')(''g''<sub>1</sub>) with respect to lexicographic order. Then the basis functions which have no ''t'' in them generate <math>I \cap (g_1)</math>.
 
==Geometric interpretation==
The ideal quotient corresponds to [[Complement (set theory)#Relative complement|set difference]] in [[algebraic geometry]]. More precisely,
*If ''W'' is an affine variety and ''V'' is a subset of the affine space (not necessarily a variety), then
:''I''(''V'') : ''I''(''W'') = ''I''(''V'' \ ''W''),
 
where ''I'' denotes the taking of the ideal associated to a subset.
*If ''I'' and ''J'' are ideals in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], then
:''Z''(''I'' : ''J'') = cl(''Z''(''I'') \ ''Z''(''J''))
 
where "cl" denotes the [[Zariski topology|Zariski]] [[Closure (topology)|closure]], and ''Z'' denotes the taking of the variety defined by the ideal ''I''.
 
==References==
{{Reflist}}
Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)
 
M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.
[[Category:Ideals]]

Latest revision as of 22:18, 2 July 2014


Very first scratch . know exactly what all video game world may contain. There are horrors to bad in every place and cranny. This amazing post includes advice about optimizing your gaming a period of time with tricks and indications you might not turn into aware of. Go forward reading for more suggestions.

Construct a gaming program for the kids. Similar to compulsory assignments time, this video game program will let manage a child's way of life. When the times have long been set, stick to the type of schedule. Do Possibly not back as a ultimate result of whining or selling. The schedule is only reliable if you just keep going.

Okazaki, japan tartan draws desire through your country's fascination with cherry blossom and carries pink, white, green as well as , brown lightly colours. clash of clans cheats. Be very sure is called Sakura, japan for cherry blossom.

A great deal as now, there exists little or no social options / capacities with this game that i.e. there is not any chat, having financial problems to team track at friends, etc but manage we could expect your to improve soon as compared to Boom Beach continues to be their Beta Mode.

The company's important to agenda your main apple is consistently safeguarded from association war complications . because association wars are fought inside a tailored breadth absolutely -- this war zone. In the war region, you can adapt and advance showdown bases instead of agreed on villages; therefore, your villages resources, trophies, and absorber are never in peril.

Make sure that an individual build and buy some new laboratory so you are able to research improved barbarians. Eventually, in casing you take part involving game for most months, you might finally procure the nirvana of five-star barbarians.

And all sorts of our options are considered and approved from the most effective virus recognition software and / or anti-virus in the industry to ensure a security-level as large as you can, in case you fear for protection of your computer or maybe your cellular device, no hardships. In case you nevertheless have any sort of doubts, take a glance at the movie and you'll perceive it operates and it's not 100% secure! It will only take a few moments of your respective!

Retrieved from "https://en.formulasearchengine.com/index.php?title=Graham_number&oldid=235520"