wiktionary Idealideal ideals Ideal may refer to tocright In philosophy Ideal ethics , values that one actively pursues as goals Platonic ideal , a philosophical idea of trueness of form, associated with Plato In mathematics Ideal ring theory , special subsets of a ring considered in abstract algebra Semigroup Subsemigroups and ideals Ideal , special subsets of a semigroup Ideal order theory , special kind of lower sets of an order Ideal set theory , a collection of sets regarded as small or negligible Ideal Lie algebra , a particular subset in lie algebra In science Ideal chain , in science, the simplest model describing a polymer Ideal gas law , in physics, governing the pressure of an ideal gas Ideal transformer, an electrical transformer having zero resistance and perfect magnetic threading Ideal Final Result , in TRIZ methodology, the best possible solution Ideality phrenology , in phrenology, the disposition towards refinement and perfection Thought experiment , sometimes called an ideal experiment Ideal type , a social science term Ideal solution , a solution with thermodynamic properties analogous to those of a mixture of ideal gases Entertainment Ideal Film Company , a leading British film studio of the Silent Era Ideal TV series Ideal TV series , a British situation comedy Ideal band , late 1990s 2000s American R&B group Ideal German band , early 1980s German rock group Ideal Toy Company , a now defunct toy company Ideal album Ideal album , the 1999 eponymous debut album by the R&B group Other uses Changhe Ideal , a city car produced by a joint venture of Changhe and Suzuki IDEAL Scholars Fund , an American scholarship program for underrepresented students Ideal, Georgia , a community in the United States iDEAL , an online payment method in the Netherlands Ideal Industries , an American manufacturer of electrical connectors and tools IDEAL Network , Idea Development ... therapies Places Ideal, Illinois Ideal, South Dakota Ideal Mini School See also Idealism ... more details
lowercase title iDEALiDEAL is an E commerce payment system Internet payment method in the Netherlands ... on the Internet using direct online transfers from their bank account. iDEAL processed 4,5 million ... and 68.8 million in 2010. ref http www.ideal.nl acceptant ?s wat&lang eng GB What is iDEAL? including numbers of transfers ref iDEAL is owned by the Dutch organization Currence , which also owns Personal identification number PIN and Chipknip . The participating banks in iDEAL are ABN AMRO , ASN Bank ... http www.ideal.nl consument ?s wat&lang eng GB What is iDEAL? including a list of participating banks ref Together these are serving the vast majority of the Dutch online banking market. Process iDEAL ... internet payments. For customers, iDEAL uses the same environment as their banks online banking ... using this payment method. This is considered an advantage to the merchants. iDEAL works as follows Merchant offers iDEAL as payment method Consumer selects iDEAL and selects his bank Consumer ... en icon Official website http www.ideal.nl nl icon http www.abnamro.nl nl prive betalen ideal introductie.html ABN AMRO on iDEAL nl icon http www.asnbank.nl index.asp?NID 10456 ASN Bank on iDEAL nl icon http www.frieslandbank.nl OnlineBankieren Pages iDEAL.aspx Friesland Bank on iDEAL nl icon http www.ing.nl particulier klantenservice veelgestelde vragen betalen betalen ideal index.aspx?faquri tcm 7 6592 ING bank on iDEAL nl icon http www.rabobank.nl particulieren producten betalen internetbankieren ideal Rabobank on iDEAL nl icon http www.regiobank.nl particulier home klantenservice internet bankieren ideal.html RegioBank on iDEAL nl icon http www.snsbank.nl particulier betalen hoe werkt ideal2.html SNS bank on iDEAL nl icon http www.triodos.nl nl particulieren service veelgestelde vragen ?faqId 425&limit false Triodos Bank on iDEAL nl icon http www.vanlanschot.nl vanlanschot productinformatie online bankieren ideal.html Van Lanschot on iDEAL nl icon References references DEFAULTSORT ... more details
In mathematics , an Ideal ring theory ideal of a commutative ring is said to be irreducible if it cannot be written as a finite intersection of ideals properly containing it. Every prime ideal is irreducible. Every irreducible ideal of a Noetherian ring is a primary ideal , and consequently for Noetherian rings an irreducible decomposition is a primary decomposition . Every primary ideal of a principal ideal domain is an irreducible ideal. Every irreducible ideal is a primal ideal . Fact date April 2009 An element of an integral domain is prime element prime if, and only if, an ideal generated by it is a nonzero prime ideal. This is not true for irreducible ideals an irreducible ideal may be generated by an element that is not an irreducible element , as is the case in math mathbb Z math for the ideal math 4 mathbb Z math It is not the intersection of two strictly greater ideals. The notion is more easily understood in terms of algebraic geometry . An ideal I of a ring A is irreducible if, and only if.the algebraic set it defines is irreducible space irreducible that is, any open subset is dense for the Zariski topology , or equivalently if the closed space of spec A consisting of prime ideals containing I is irreducible for the affine scheme spectral topology . If k is an algebraically closed field , choosing the radical of an ideal radical of an irreducible ideal of a polynomial ring over k is the same thing as choosing an embedding of the affine variety of its Nullstelle in the affine space. See also Laskerian ring , irreducible module , irreducible space Category Ring theory Category Algebraic topology Abstract algebra stub ... more details
In mathematics , an element a of a commutative ring A is called relatively prime to an ideal Q if whenever ab is an element of Q then b is also an element of Q . A proper Ideal ring theory ideal Q of a commutative ring A is said to be primal if the elements that are not prime to it form an ideal. References http www.ams.org journals proc 1950 001 01 S0002 9939 1950 0032584 8 S0002 9939 1950 0032584 8.pdf On primal ideals , Ladislas Fuchs Category Commutative algebra ... more details
Commented out because image was deleted Image Ideal logo.JPG thumb right 300px The logo of ideal bike company Too much Image Ideal bike2.JPG thumb right The front of a Ideal mountain bike. Ideal Bikes is a Greece Greek bicycle manufacturer located in Agios Vassileios , near Patras . History The company Maniatopoulos Bros S.A. was founded in Patras in 1926, originally as an importer of bicycles and later motorbikes. In 1987 it was renamed Nikos Maniatopoulos S.A. The brand Ideal is used since 1977 in 1991 production moved to its present modern factory in Agios Vassileios , while an aggressive export campaign established the brand in over 20 countries. The company maintains very advanced Design, Development and production technology infrastructure and Ideal Bikes is an extremely popular brand, especially in Europe, with a network of distributors throughout the continent. Other branches of the Maniatopoulos family have been involved in the automotive sector, like the owners of MAVA the Renault importer in Greece, who designed and produced the MAVA Renault Farma automobile . Teams In its first year as a professional team the Team Volksbank started as Volksbank Ideal . The swiss lettonian team Rietumu Bank Riga is sponsored by Ideal and begun as Rietumu Bank Riga Ideal . Several mountain bike teams have won international titles on Ideal bikes ref See also Teams on http www.idealcyklar.se ref Ideal Bikes Team Team Emmedue Team Zwillingscraft Team Biking Cyprus References Reflist External links http www.idealbikes.net http www.idealbikesteam.com The Ideal Bikes Team based in Varese Italy Category Patras Category Cycle manufacturers of Greece Category Companies established in 1926 Category Mountain bike manufacturers cycle manufacturer stub de Ideal Bikes el Ideal Bikes nl Ideal Bikes fi Ideal Bikes ... more details
Orphan date September 2008 Unreferenced date March 2007 An ideal moment in physics mechanics is a Moment physics moment which is not associated with a force . DEFAULTSORT Ideal Moment Category Solid mechanics engineering stub ... more details
In mathematics , especially ring theory , a regular ideal can refer to multiple concepts. In operator theory , a right ideal ring theory ideal math mathfrak i math in a possibly non unital ring A is said ... for every math x in A math . harv Jacobson 1956 In commutative algebra a regular ideal refers to an ideal ... . ref . harv Larsen McCarthy 1971 loc p.42 This article will use regular element ideal to help distinguish this type of ideal. A two sided ideal math mathfrak i math of a ring R can also be called a von Neumann regular ideal if for each element x of math mathfrak i math there exists a y in math ..., regular ideal has been used to refer to an ideal J of a ring R such that the quotient ring R J is von ... of regular ideal. Since the adjective regular has been overloaded, this article adopts the alternative .... A two sided ideal math mathfrak i math is modular if and only if math A mathfrak i math is unital. In a unital ring, every ideal is modular since choosing e 1 works for any right ideal. So, the notion ... to see that an ideal containing a modular ideal is itself modular. Somewhat surprisingly, it is possible to prove that even in rings without identity, a modular right ideal is contained in a maximal right ideal sfn Jacobson 1956 loc p.6 . The intersection of all maximal right ideals which are modular ... element in M , then the annihilator of x is a regular maximal right ideal in A . If A is a ring without maximal right ideals, then A cannot have even a single modular right ideal. Regular element ideals Every ring with unity has at least one regular element ideal the trivial ideal R itself ... is again a regular element ideal. Clearly any ideal containing a regular element ideal is again a regular element ideal. Examples In an integral domain , every nonzero element is a regular element, and so every nonzero ideal is a regular element ideal. The nilradical of a ring nilradical of a commutative .... This gives an example of an ideal which is not a regular element ideal. In an Artinian ring , each ... more details
In mathematics , a tertiary ideal is an two sided Ideal ring theory ideal in a perhaps noncommutative Ring mathematics ring that cannot be expressed as a nontrivial intersection of a right fractional ideal with another ideal. Tertiary ideals generalize primary ideal s to the case of noncommutative ring s. Although primary decomposition s do not exist in general for ideals in noncommutative rings, tertiary decompositions do, at least if the ring is Noetherian ring Noetherian . Every primary ideal is tertiary. Tertiary ideals and primary ideals coincide for commutatitve rings. To any two sided ideal, a tertiary ideal can be associated called the tertiary radical, defined as math t I r in R mbox mbox forall s notin I, mbox exists x in s mbox x notin I text and x r subset I . , math Then t I always contains I . If R is a not necessarily commutative Noetherian ring and I a right ideal in R , then I has a unique irredundant decomposition into tertiary ideals math I T 1 cap dots cap T n math . See also Primary ideal Lasker Noether theorem References Citation last Riley first J.A. title Axiomatic primary and tertiary decomposition theory journal Trans. Amer. Math. Soc. year 1962 volume 105 pages 117 201 http www.encyclopediaofmath.org index.php Tertiary ideal Tertiary ideal , Encyclopedia of Mathematics, Springer Online Reference Works. Citation last Behrens first Ernst August title Ring Theory publisher Verlag Academic Press year 1972 url http books.google.ch books?id ZKGq4IQHhHUC&lpg PP1&pg PP1 v onepage&q &f false Citation last kurata first yoshiki title On an additive ideal theory in a non associative ring journal Mathematische Zeitschrift volume 88 issue 2 year 1965 doi 10.1007 BF01112095 pages 129 135 url http www.springerlink.com content h772w68514700345 algebra stub DEFAULTSORT Tertiary ideals Category Algebra ... more details
Infobox football club clubname Ideal SC image fullname Ideal Sports Club nickname Ideal founded ground capacity chrtitle chairman mgrtitle Manager manager league Montserrat Championship season 2004 Montserrat Championship 2004 position 1st current website Ideal Sports Club is a Montserrat Montserratian sporting club based in Brades . Ideal SC is best known for their association football football team, whom plays in the Montserrat Championship the highest tier of football on the island. The football club are the most recent champions of the league, although the last competitive match took place in 2004. ref cite web last Sch ggl first Hans title Montserrat List of Champions url http www.rsssf.com tablesm montserratchamp.html publisher Rec.Sport.Soccer Statistics Foundation accessdate 4 August 2011 date 29 December 2004 ref Caribbean footyclub stub See Also Montserrat Championship References reflist DEFAULTSORT Ideal Sc Category Football clubs in Montserrat Category Caribbean football club stubs es Ideal SC ... more details
Image Ideal circles.GIF thumb right 200px Three ideal triangles in the Poincar disk model Image IdealTriangle HalfPlane.jpg thumb right 200px Two ideal triangles in the upper half plane model In hyperbolic geometry an ideal triangle is a hyperbolic triangle Hyperbolic geometry hyperbolic triangle whose three vertices all lie on the circle at infinity. In the hyperbolic metric, any two ideal triangles are Congruence geometry congruent . Ideal triangles are also sometimes called triply asymptotic triangles or trebly asymptotic triangles . Models In the Poincar disk model of the hyperbolic plane, an ideal triangle is bounded by three circles which intersect the boundary circle at right angles. In the Poincar half plane model , an ideal triangle is modeled by an arbelos , the figure between three mutually tangent semicircle s. And in the Beltrami Klein model of the hyperbolic plane, an ideal triangle is modeled by a Euclidean triangle that is circumscribed by the boundary circle. Note that in the Beltrami Klein model, the angles at the vertices of an ideal triangle are not zero, because the Beltrami Klein model, unlike the Poincar disk and half plane models, is not conformal map conformal ... 1 at every point The interior angles of an ideal triangle are all zero. Any ideal triangle has area &pi . Any ideal triangle has infinite perimeter. The inscribed circle to an ideal triangle ... triangles more generally in a non ideal triangle, the distance to the second closest side is strictly ... be multiplied by 1 K and the lengths and distances should be multiplied by 1 &radic K . Real ideal triangle group Image Ideal triangle hyperbolic tiling.svg thumb right The Poincar disk model tiled with ideal triangles. The real ideal triangle group is the reflection group generated by reflections of the hyperbolic plane through the sides of an ideal triangle. Algebraically, it is isomorphic ... author Schwartz, Richard Evan title Ideal triangle groups, dented tori, and numerical analysis ... more details
Infobox Company company name Ideal Industries company logo Image Ideal Industries logo.png 180px company type Privately owned foundation 1916 in Chicago, Illinois founder J. Walter Becker location Sycamore, Illinois key people num employees industry Manufacturing revenue products Connectors, hand tools, testers, meters homepage http www.idealindustries.com www.idealindustries.com Ideal Industries is an United ... and telecommunications industries. ref cite web author Ideal Industries title About Us url http www.idealindustries.com about accessdate 2010 03 26 ref History Ideal was founded in 1916 by J. Walter Becker as the Ideal Commutator Dresser Company in Chicago, Illinois , manufacturing commutator electric commutator grinding dresser dresser stones. ref cite web author Ideal Industries title ..., Becker relocated the company to its current location in Sycamore, Illinois . ref cite web author Ideal Industries title Recruiting brochure url http www.idealindustries.com media pdfs about ideal recruiting brochure.pdf accessdate 2010 03 27 ref In 2010, Ideal acquired Western Forge ref cite web work Fabricating & Metalworking title Ideal Industries Acquires Western Forge url http www.fandmmag.com web online Industry News Ideal Industries Acquires Western Forge 1 4115 accessdate 2010 03 17 ref and Pratt Read . ref cite press release publisher Ideal Industries title IDEAL Acquires Tool Manufacturer Pratt Read url http www.pitchengine.com ideal acquires tool manufacturer pratt read 53194 date ... 25, 2010, Ideal would acquire SK Hand Tools . ref cite web last Yue first Lorene work Crain Communications Chicago Business title SK Hand Tool assets sold to Ideal Industries url http www.chicagobusiness.com article 20100823 NEWS05 100829965 sk hand tool assets sold to ideal industries date 2010 08 23 accessdate 2010 10 06 ref ref cite press release publisher Ideal Industries title IDEAL Acquires ... links http www.idealindustries.com Ideal Industries web site Tool manufacturers Category Electrical ... more details
Unreferenced date December 2009 An ideal is a principle or Value personal and cultural value that one ... to have an ideal of honesty but is willing to lie to protect a friend is demonstrating that not only does he hold friendship as an ideal, but, that it is a more important one than honesty. Thus ideals ... between the ideal of truth , which would advise hearing out all evidence, and the ideal of fairness ... power to continue to pursue them. Idols and heroes A different form of ideal is an idolatry idol ..., it is too complex and multi faceted to be considered an ideal in the abstract sense. However, when ..., the life of Muhammad is held up as ideal , but must be interpreted for believers through the tale of his life, or Sirah Rasul Allah sira , and his many sayings, the hadith . Ideal and virtue Given the complexity ..., is an ideal that one can make a habituation habit . Relative ideal In formal axiology, Robert S. Hartman contended that being ideal means that something is the best member of the set of all things of that class. For example, the ideal student is the best member of the set of all students in exactly the same way that the ideal circle is the best circle that can be imagined of the class of all circles. Since we can define the properties that the ideal member of a class should have, the value of any actual object can be empirically determined by comparing it to the ideal. The closer an object s actual properties match up to the properties of the ideal, the better the object is. For example ... in general, each particular object ought to become more like its ideal. In ethics , by analogy, each person should attempt to become more of an ideal person, and a person s morality can actually be measured by examining how close they live up to their ideal self. See also Idealism DEFAULTSORT Ideal Ethics Category Core issues in ethics Category Philosophy of life be x old bg da Ideal de Ideal Philosophie el es Ideal filosof a hr Ideal ka kk nl Ideaal ... more details
In mathematics , specifically commutative algebra , a proper Ideal ring theory ideal Q of a commutative ... of Q , for some n 0 . For example, in the ring of integers Z , p sup n sup is a primary ideal ... every ideal of a Noetherian ring has a primary decomposition , that is, can be written as an intersection ..., ref To be precise, one usually uses this fact to prove the theorem. ref an irreducible ideal ... are assumed to be commutative rings with identity. Examples and properties Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime ideal semiprime . Every primary ideal is primal ideal primal . ref For the proof of the second part see the article of Fuchs ref If Q is a primary ideal, then the Radical of an ideal radical of Q is necessarily a prime ideal P , and this ideal is called the associated prime ideal of Q . In this situation, Q is said to be P primary . If P is a maximal prime ideal, then any ideal containing a power of P is P primary. Not all P primary ideals need be powers of P for example the ideal x , y sup 2 sup is P primary for the ideal P x , y in the ring k x , y , but is not a power of P . In general powers of a prime ideal P need not be P primary. An example is given by taking R to be the ring k x , y , z xy &minus z sup 2 sup , with P the prime ideal x , z . If Q ... , so Q is not P primary. However every ideal Q with radical P is contained in a smallest P primary ideal, consisting of all elements a such that ax is in Q for some x not in P . In particular there is a smallest P primary ideal containing P sup n sup , called the n th symbolic power of P . If A is a Noetherian ring and P a prime ideal, then the kernel of math A to A P math , the map from A to the localization ... 1963 pages 119 MR 0155861 External links http www.encyclopediaofmath.org index.php Primary ideal Primary ideal at Encyclopaedia of Mathematics Category Commutative algebra Category Ideals eo Unuagrada ... more details
Orphan date October 2010 In mathematics the Jacobian ideal or gradient ideal is the Ideal ring theory ideal generated by the Jacobian matrix and determinant Jacobian of a function or germ mathematics function germ . Let math mathcal O x 1, ldots,x n math denote the ring mathematics ring of smooth functions and f a function in the ring. The Jacobian ideal of f is math J f left langle frac partial f partial x 1 , ldots, frac partial f partial x n right rangle. math See also Milnor number Unfolding Category Singularity theory Category Ideals Abstract algebra stub ... more details
The ego ideal is the inner image of oneself as one wants to become. ref Salman Akhtar , Comprehensive Dictionary of Psychoanalysis 2009 p. 89 ref Alternatively, The Freud ian notion of a perfect or ideal ... after certain people whom...he regards as ideal . ref Eric Berne, A Layman s Guide to Psychiatry and Psychoanalysis ... ideal or ideal ego has been defined as an image of the perfect self towards which the Id, ego and super ego Ego ego should aspire. ref Janine Chasseguet Smirgel, The Ego Ideal A Psychoanalytic Essay on the Malady of the Ideal , 1st American ed., trans. Paul Barrows, introduction by Christopher ... 0 393 01971 3. ref Freud, ego ideal, and superego In Freud s On Narcissism On Narcissism an Introduction ... ideal and of the self observing agency related to it, which were the basis of what was ultimately ... , in Sigmund Freud, On Metapsychology PFL 11 p. 62 ref Freud considered that the ego ideal was the heir to the narcissism of childhood the ideal ego is now the target of the self love which was enjoyed ... ideal ...heir to the original narcissism in which the childish ego enjoyed self sufficiency ... forms of love choice...the object serves as a substitute for some unattained ego ideal of our own , and further suggested that in group formation the group ideal...governs the ego in the place of the ego ideal . ref Freud, Civilization p. 143 and p. 160 ref With The Ego and the Id 1923 , however ... in the ego, a differentiation in the ego, which may be called the ego ideal or super ego , ref Freud ... it, the ego ideal disappears almost completely as a technical term ref Richards, p. 348 ref for Freud ......the vehicle of the ego ideal by which the ego measures itself...precipitate of the old picture ... . ref Sigmund Freud, New Introductory Lectures on Psychoanalysis PFL 2 p. 96 ref Stekel s ego ideal Ernest Jones records that I once asked Freud if he regarded an ego ideal as a universal attribute, and he replied with a puzzled expression Do you think Wilhelm Stekel Stekel has an ego ideal? . ref ... more details
In ring theory , a branch of abstract algebra , a principal ideal is an ideal ring theory ideal I in a ring mathematics ring R that is generated by a single element a of R . More specifically a left principal ideal of R is a subset of R of the form Ra ra r in R a right principal ideal is a subset of the form aR ar r in R a two sided principal ideal is a subset of the form RaR r sub 1 sub as sub 1 sub ... r sub n sub as sub n sub r sub 1 sub , s sub 1 sub ,..., r sub n sub , s sub n sub in R . If R ... to write the ideal generated by a as a . Not all ideals are principal. For example, consider the commutative ... number complex coefficients. The ideal x , y generated by x and y , which consists of all the polynomials ... . A ring in which every ideal is principal is called principal , or a principal ideal ring . A principal ideal domain PID is an integral domain that is principal. Any PID must be a unique factorization ... greatest common divisor s may be used to find a generator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest common divisors of elements ... of the ideal a , b . For a Dedekind domain R , we may also ask, given a non principal ideal I of R , whether there is some extension S of R such that the ideal of S generated by I is principal said ... ideal theorem principal ideal theorem of class field theory states that every integer ring ... that every ideal of R becomes a principal ideal of S . In this theorem we may take S to be the ring ... of the fraction field of R , and this is uniquely determined by R . Krull s principal ideal theorem states that if R is a Noetherian ring and I is a principal, proper ideal of R , then I has height ... algebra ca Ideal principal cs Hlavn ide l teorie okruh de Hauptideal es Ideal principal fr Id al principal nl Hoofdideaal pl Idea g wny pt Ideal principal ru sv Principalideal uk ... more details
In mathematics , more specifically ring theory , a left, right or two sided Ideal ring theory ideal of a Ring mathematics ring is said to be a nil ideal if each of its elements is Nilpotent element nilpotent .... 13 ref The Nilradical of a ring nilradical of a commutative ring is an example of a nil ideal in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative ring s. Nil ideals are still ... In a commutative ring, the set of all nilpotent element s forms an ideal known as the nilradical of the ring. Therefore, an ideal of a commutative ring is nil if, and only if, it is a subset of the nilradical that is, the nilradical is the ideal maximal with respect to the property that each of its ... of a commutative ring R , a R is an ideal that is in fact nil. This is because any element of the principal ideal generated by a is of the form a r for r in R , and if a sup n sup 0, a r sup n sup a sup n sup r sup n sup 0. It is not in general true however, that a R is a nil one sided ideal in a noncommutative ... a maximal nil ideal the nilradical of the ring. The existence of such a maximal nil ideal ... nil. However, the truth of the assertion that the sum of two left nil ideals is again a left nil ideal ... to nilpotent ideals The notion of a nil ideal has a deep connection with that of a nilpotent ideal , and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course ... be avoided for a nil ideal to qualify as nilpotent. In a Artinian ring right artinian ring , any nil ideal is nilpotent. sfn Isaacs year 1993 loc Corollary 14.3, p. 195 This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal due to the artinian hypothesis , the result follows. In fact, this has been generalized ... . See also K the conjecture Nilpotent ideal Nilradical Jacobson radical Notes reflist 2 References ... more details
In mathematics, a Frink ideal , introduced by Orrin Frink, is a certain kind of subset of a partially ordered set . Basic definitions LU A is the set of all lower bound s of the set of all upper bound s of the subset A of a partially ordered set . A subset I of a partially ordered set P , is a Frink ideal , if the following condition holds For every finite subset S of P , S math subseteq math I implies that LU S math subseteq math I . A subset I of a partially ordered set P , is a normal ideal or a cut if LU I math subseteq math I . Remarks Every Frink ideal I is a lower set . A subset I of a lattice P , is a Frink ideal if and only if it is a lower set that is closed under finite joins suprema . Every normal ideal is a Frink ideal. Related notions pseudoideal Doyle pseudoideal directed ideal References Frink, Orrin, Ideals in Partially Ordered Sets . American Mathematical Monthly , 61 1954 , 223 234. Niederle, Josef, Ideals in ordered sets . Rendiconti Circ. Math. Palermo , 55 2006 , 287 295. Category Order theory ... more details
In mathematics , more specifically in ring theory , a maximal ideal is an ideal ring theory ideal which ... in Mathematics year 2002 isbn 0 387 95385 X ref In other words, I is a maximal ideal of a ring R if there are no other ... ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one sided maximal ideal A is not necessarily two sided, the quotient R A is not necessarily a ring, but it is a simple module over R . If R has a unique maximal right ideal, then R is known as a local ring , and the maximal right ideal is also the unique maximal left and unique maximal two sided ideal of the ring, and is in fact the Jacobson radical J R . It is possible for a ring to have a unique maximal ideal and yet lack unique maximal one sided ideals for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal ideal, but there are many maximal ... sided and maximal two sided ideals. Given a ring R and a proper ideal I of R that is I R , I is a maximal ideal of R if any of the following equivalent conditions hold There exists no other proper ideal J of R so that I J . For any ideal J with I J , either J I or J R . The quotient ring R I is a simple ... will be given. For a right ideal A of a ring R , the following conditions are equivalent to A being a maximal right ideal of R There exists no other proper right ideal B of R so that A B . For any right ideal B with A B , either B A or B R . The quotient module R A is a simple right R module. Maximal right left two sided ideals are the duality mathematics dual notion to that of minimal ideal s. Examples In the ring Z of integers the maximal ideals are the principal ideal s generated by a prime number. More generally, all nonzero prime ideal s are maximal in a principal ideal domain . The maximal ... field K are the ideal of the form nowrap x sub 1 sub &minus a sub 1 sub ,..., x sub n sub ... more details
Refimprove date December 2009 In mathematics In mathematics , ideal theory is the theory of ideal ring theory ideal s in commutative ring s and is the precursor name for the contemporary subject of commutative algebra . The name grew out of the central considerations, such as the Lasker Noether theorem in algebraic geometry , and the ideal class group in algebraic number theory , of the commutative algebra of the first quarter of the twentieth century. It was used in the influential Bartel Leendert van der Waerden van der Waerden text on abstract algebra from around 1930. The ideal theory in question had been based on elimination theory , but in line with David Hilbert s taste moved away from algorithmic methods. Gr bner basis theory has now reversed the trend, for computer algebra . The importance of the ideal in general of a module mathematics module , more general than an ideal , probably led to the perception that ideal theory was too narrow a description. Valuation theory , too, was an important technical extension, and was used by Helmut Hasse and Oscar Zariski . Bourbaki used commutative algebra sometimes local algebra is applied to the theory of local ring s. D. G. Northcott s 1953 Cambridge Tract Ideal Theory reissued 2004 under the same title was one of the final appearances of the name. In political philosophy In political philosophy , ideal theory refers to argument concerning political or social arrangements under favorable assumptions. The phrase is associated with the work of John Rawls . ref name stanford rawls http plato.stanford.edu entries rawls IdeNonIdeThe Stanford Encyclopedia of Philosophy , article on Rawls ref References reflist DEFAULTSORT Ideal Theory Category History of mathematics Category Ideals Category Commutative algebra Category Political philosophy nl Ideaaltheorie ... more details
Distinguish primary ideal In mathematics , a left primitive ideal in ring theory is the Annihilator ring theory annihilator of a simple module simple left module mathematics module . A right primitive ideal is defined similarly. Note that despite the name left and right primitive ideals are always two sided ideals. The quotient of a ring by a left primitive ideal is a left primitive ring . References Citation last1 Isaacs first1 I. Martin title Algebra publisher Brooks Cole Publishing Company isbn 0 534 19002 2 year 1994 Category Ideals Category Module theory Abstract algebra stub bg ... more details
about ideals in ring theory prime ideals in order theory ideal order theory Prime ideals In algebra which is a branch of mathematics , a prime ideal is a subset of a ring mathematics ring which shares ... of a given prime number or zero. Primitive ideal s are prime, and prime ideals are both primary ideal primary and semiprime ideal semiprime . Prime ideals for commutative rings An ideal ring theory ideal P of a commutative ring R is prime if it has the following two properties If a and b are two ... can therefore say A positive integer n is a prime number if and only if the ideal n Z is a prime ideal ... complex coefficients, then the ideal generated by the polynomial Y sup 2 sup &minus X sup 3 sup &minus X &minus 1 is a prime ideal see elliptic curve . In the ring Z X of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even. In any ring R , a maximal ideal is an ideal M that is maximal ... of R , namely M itself and the entire ring R . Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general. If M is a smooth ... smooth functions f with f x 0 forms a prime ideal even a maximal ideal in R . Properties An ideal ..., a commutative ring is an integral domain if and only if 0 is a prime ideal. An ideal I is prime if and only ... one prime ideal in fact it contains at least one maximal ideal which is a direct consequence of Krull ... of the spectrum. The preimage of a prime ideal under a ring homomorphism is a prime ideal ... proper ideal is prime, then the ring is a field. If the ideal 0 is prime, then the ring R is an integral domain. If q is any non zero element of R and the ideal math q 2 math is prime, then it contains q and then q is invertible. A nonzero principal ideal is prime if and only if it is generated by a prime element . In a UFD, every nonzero prime ideal contains a prime element. Uses One use of prime ... more details
About shopping channel the Chinese manhua An Ideal World Multiple issues refimprove February 2011 COI February 2011 wikify February 2011 Infobox TV channel name Ideal World logosize 250px logofile Ideal World logo.png logoalt launch 16 April 2000 picture format 576i SDTV owner Ideal Shopping Direct Ltd country United Kingdom web http www.idealworld.tv idealworld.tv sister names Ideal Extra, br Ideal & More, br Create and Craft terr serv 1 Freeview UK Freeview terr chan 1 Channel 22 small Ideal World ... 644 small Ideal World small br Channel 651 small Ideal Extra small br Channel 668 small Ideal & More ... Ideal Extra small br Channel 812 small Ideal World small br Channel 813 small Create and Craft small cable serv 1 Virgin Media cable chan 1 Channel 747 small Ideal World small cable serv 2 Smallworld Cable cable chan 2 Channel 230 small Create and Craft small br Channel 231 small Ideal World small br Channel 232 small Ideal Extra small online serv 1 TVCatchup online chan 1 http www.tvcatchup.com watch.html?c 39 Watch live UK only small Ideal World small Ideal World is a infomercial shopping channel in the United Kingdom UK whose sister shopping channels include Create and Craft , Ideal Extra and Ideal & More . It is owned by Ideal Shopping Direct Plc. ref cite web url http www.ofcom.org.uk static tvlicensing dtt 947.htm title Licence details for Ideal World accessdate 2007 06 06 publisher Ofcom ref Their sister health and beauty channel, Vitality, closed early in 2008. Ideal Shopping ... Bargains.com. The channel broadcasts from its studios in Peterborough . Ideal World currently broadcasts .... The company later produced its own catalogues, inserted into the Daily Mail , and sold at the Ideal ... as live TV studios, offices, warehouse, and call centre. Ideal World was launched on 17 April 2000 ... http www.memorabletv.com documentary t tellysalesahomeshoppingstorystory.htm Telly Sales . Ideal s motto ... until September 2002. Ideal World has not disclosed whether a cause was found for the fire. As the television ... more details
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In mathematics , more specifically ring theory , an Ideal ring theory ideal , I , of a Ring mathematics ring is said to be a nilpotent ideal , if there exists a natural number k such that I sup k sup 0. sfn Isaacs year 1993 loc p. 194 By I sup k sup , it is meant the additive subgroup generated by the set of all products of k elements in I . sfn Isaacs year 1993 loc p. 194 Therefore, I is nilpotent if and only if there is a natural number k such that the product of any k elements of I is 0. The notion of a nilpotent ideal is much stronger than that of a nil ideal in many classes of rings. There are, however, instances when the two notions coincide this is exemplified by Levitzky s theorem . ref Isaacs, Theorem 14.38, p. 210 ref sfn Herstein year 1968 loc Theorem 1.4.5, p. 37 The notion of a nilpotent ideal, although interesting in the case of commutative ring s, is most interesting in the case of noncommutative ring s. Relation to nil ideals The notion of a nil ideal has a deep connection with that of a nilpotent ideal, and in some classes of rings, the two notions coincide. If an ideal is nilpotent, it is of course nil, but a nil ideal need not be nilpotent for more reason than one. The first is that there need not be a global upper bound on the exponent required to annihilate various elements of the nil ideal, and secondly, each element being nilpotent does not force products of distinct elements to vanish. sfn Isaacs year 1993 loc p. 194 In a right artinian ring, any nil ideal is nilpotent. ref Isaacs, Corollary 14.3, p. 195 ref This is proven by observing that any nil ideal is contained in the Jacobson radical of the ring, and since the Jacobson radical is a nilpotent ideal due to the artinian hypothesis , the result follows. In fact, this can be generalized to right noetherian rings this result is known as Levitzky s theorem . sfn Herstein year 1968 loc Theorem 1.4.5, p. 37 See also K the conjecture Nilpotent element Nil ideal Nilradical Jacobson radical Notes reflist ... more details
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