See also: Ideal, ideał, ideál, and idéal

EnglishEdit

 
English Wikipedia has an article on:
Wikipedia

EtymologyEdit

From French idéal, from Late Latin ideālis (existing in idea), from Latin idea (idea); see idea.
In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

PronunciationEdit

  • Rhymes: -iːəl
  • IPA(key): /aɪˈdɪəl/, /aɪˈdiː.əl/
  • (file)

AdjectiveEdit

ideal (comparative more ideal, superlative most ideal)

  1. Optimal; being the best possibility.
  2. Perfect, flawless, having no defects.
    • 1751 April 13, Samuel Johnson, The Rambler, Number 112, reprinted in 1825, The Works of Samuel Johnson, LL. D., Volume 1, Jones & Company, page 194,
      There will always be a wide interval between practical and ideal excellence; [] .
  3. Pertaining to ideas, or to a given idea.
  4. Existing only in the mind; conceptual, imaginary.
    • 1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:
      The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
    • 1818, [Mary Shelley], “Chapter 4”, in Frankenstein; or, The Modern Prometheus. [], volume (please specify |volume=I to III), London: [] [Macdonald and Son] for Lackington, Hughes, Harding, Mavor, & Jones, OCLC 830979744:
      Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
    • 1836 March – 1837 October, Charles Dickens, “(please specify the chapter name)”, in The Posthumous Papers of the Pickwick Club, London: Chapman and Hall, [], published 1837, OCLC 28228280:
      At first, he began to doubt the reality of his adventures, but the acute pain in his shoulders when he attempted to rise, assured him that the kicking of the goblins was certainly not ideal.
  5. Teaching or relating to the doctrine of idealism.
    the ideal theory or philosophy
  6. (mathematics) Not actually present, but considered as present when limits at infinity are included.
    ideal point
    An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

SynonymsEdit

Derived termsEdit

Related termsEdit

TranslationsEdit

The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

NounEdit

ideal (plural ideals)

  1. A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
    Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz
  2. (algebra, ring theory) A subring closed under multiplication by its containing ring.
    Let   be the ring of integers and let   be its ideal of even integers. Then the quotient ring   is a Boolean ring.
    The product of two ideals   and   is an ideal   which is a subset of the intersection of   and  . This should help to understand why maximal ideals are prime ideals. Likewise, the union of   and   is a subset of  .
    • 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd Edition, Cambridge University Press, page 47,
      In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.
    • 2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45,
      If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.
    • 2010, W. D. Burgess, A. Lashgari, A. Mojiri, Elements of Minimal Prime Ideals in General Rings, Sergio R. López-Permouth, Dinh Van Huynh (editors), Advances in Ring Theory, Springer (Birkhäuser), page 69,
      However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.
  3. (algebra, order theory, lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).[1]
    • 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,
      An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.
    • 2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125,
      1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
    • 2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186,
      Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.
  4. (set theory) A collection of sets, considered small or negligible, such that every subset of each member and the union of any two members are also members of the collection.
    Formally, an ideal   of a given set   is a nonempty subset of the powerset   such that:  ,   and  .
  5. (algebra, Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket [𝖌,𝖍] is a subset of 𝖍.
    • 1975, Che-Young Lee (translator), Zhe-Xian Wan, Lie Algebras, Pergamon Press, page 13,
      If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
    • 2006, W. McGovern, The work of Anthony Joseph in classical representation theory, Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov (editors), Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3,
      What really put primitive ideals in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such ideal is the annihilator of a very special kind of simple module, namely a highest weight module.
    • 2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73,
      Next let   be an arbitrary semisimple Lie algebra. Then   can be written uniquely as a direct sum   of simple ideals (Theorem 5.2).
  6. (algebra) A subsemigroup with the property that if any semigroup element outside of it is added to any one of its members, the result must lie outside of it.[2]
    The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.

SynonymsEdit

AntonymsEdit

HyponymsEdit

Derived termsEdit

TranslationsEdit

ReferencesEdit

  1. ^   Boolean prime ideal theorem#Prime ideal theorems on Wikipedia.Wikipedia
  2. ^ Vaughan Pratt (2004) Chapter 1 : Lattice Theory[1], boole.stanford.edu, §1.3.5

Further readingEdit

AnagramsEdit

AsturianEdit

EtymologyEdit

From Latin ideālis.

PronunciationEdit

  • IPA(key): /ideˈal/, [i.ð̞eˈal]

AdjectiveEdit

ideal (epicene, plural ideales)

  1. ideal

NounEdit

ideal m (plural ideales)

  1. ideal

CatalanEdit

EtymologyEdit

From Latin ideālis.

PronunciationEdit

AdjectiveEdit

ideal (masculine and feminine plural ideals)

  1. ideal

Derived termsEdit

NounEdit

ideal m (plural ideals)

  1. ideal

GalicianEdit

EtymologyEdit

From Latin ideālis.

PronunciationEdit

  This entry needs pronunciation information. If you are familiar with the IPA then please add some!

AdjectiveEdit

ideal m or f (plural ideais)

  1. ideal

Derived termsEdit

NounEdit

ideal m (plural ideais)

  1. ideal

GermanEdit

EtymologyEdit

Borrowed from Late Latin ideālis (existing in idea), from Latin idea (idea). Doublet of ideell.

PronunciationEdit

AdjectiveEdit

ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten)

  1. ideal (optimal, perfect)

DeclensionEdit

Derived termsEdit

Further readingEdit

  • ideal” in Duden online
  • ideal” in Digitales Wörterbuch der deutschen Sprache

IndonesianEdit

 
Indonesian Wikipedia has an article on:
Wikipedia id

EtymologyEdit

From Dutch ideaal, from French idéal, from Medieval Latin ideālis.

PronunciationEdit

  • IPA(key): [ˈɪdeal]
  • Hyphenation: idé‧al

AdjectiveEdit

ideal

  1. ideal:
    1. optimal; being the best possibility.
    2. pertaining to ideas, or to a given idea.

NounEdit

ideal (plural ideal-ideal, first-person possessive idealku, second-person possessive idealmu, third-person possessive idealnya)

  1. (mathematics) ideal: a subring closed under multiplication by its containing ring.

Alternative formsEdit

Affixed termsEdit

Related termsEdit

Further readingEdit

LuxembourgishEdit

PronunciationEdit

  • IPA(key): /ideˈaːl/, [idəˈaːl], /idiˈaːl/

AdjectiveEdit

ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)

  1. ideal

DeclensionEdit

Norwegian BokmålEdit

EtymologyEdit

From French idéal, from Late Latin ideālis (existing in idea), from Latin idea (idea).

NounEdit

ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene)

  1. ideal

ReferencesEdit

Norwegian NynorskEdit

EtymologyEdit

From French idéal, from Late Latin ideālis (existing in idea), from Latin idea (idea).

NounEdit

ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)

  1. ideal

ReferencesEdit

PiedmonteseEdit

PronunciationEdit

AdjectiveEdit

ideal

  1. ideal

NounEdit

ideal m

  1. ideal

PortugueseEdit

EtymologyEdit

Borrowed from Latin ideālis.

PronunciationEdit

  • Rhymes: -al, -aw
  • Hyphenation: i‧de‧al

AdjectiveEdit

ideal m or f (plural ideais)

  1. ideal
  2. notional

Derived termsEdit

NounEdit

ideal m (plural ideais)

  1. ideal
  2. fantasy

RomanianEdit

EtymologyEdit

From French idéal, from Latin idealis.

AdjectiveEdit

ideal m or n (feminine singular ideală, masculine plural ideali, feminine and neuter plural ideale)

  1. ideal

DeclensionEdit

Related termsEdit

Serbo-CroatianEdit

PronunciationEdit

  • IPA(key): /iděaːl/
  • Hyphenation: i‧de‧al

NounEdit

idèāl m (Cyrillic spelling идѐа̄л)

  1. ideal

DeclensionEdit

SpanishEdit

EtymologyEdit

From Latin ideālis.

PronunciationEdit

  • IPA(key): /ideˈal/ [i.ð̞eˈal]
  • Rhymes: -al
  • Syllabification: i‧de‧al

AdjectiveEdit

ideal (plural ideales)

  1. ideal

Derived termsEdit

NounEdit

ideal m (plural ideales)

  1. ideal

Further readingEdit

SwedishEdit

PronunciationEdit

NounEdit

ideal n

  1. ideal; perfect standard
  2. (mathematics) ideal; special subsets of a ring

DeclensionEdit

Declension of ideal 
Singular Plural
Indefinite Definite Indefinite Definite
Nominative ideal idealet ideal idealen
Genitive ideals idealets ideals idealens

AnagramsEdit

TagalogEdit

Alternative formsEdit

EtymologyEdit

Borrowed from Spanish ideal.

PronunciationEdit

  • Hyphenation: i‧de‧al
  • IPA(key): /ʔideˈjal/, [ʔɪ.deˈjal]

NounEdit

ideál

  1. ideal (model to be imitated)
    Synonyms: huwaran, uliran
  2. ambition
    Synonyms: mithi, mithiin, pangarap, ambisyon

Related termsEdit

AdjectiveEdit

ideál

  1. ideal (just as what one wants)
  2. worthy of imitation
    Synonym: uliran

TurkishEdit

EtymologyEdit

Borrowed from French idéal.

PronunciationEdit

  • IPA(key): /ideˈɑl/
  • Hyphenation: i‧de‧al

AdjectiveEdit

ideal

  1. ideal

NounEdit

ideal (definite accusative ideali, plural idealler)

  1. ideal

DeclensionEdit

Inflection
Nominative ideal
Definite accusative ideali
Singular Plural
Nominative ideal idealler
Definite accusative ideali idealleri
Dative ideale ideallere
Locative idealde ideallerde
Ablative idealden ideallerden
Genitive idealin ideallerin
Possessive forms
Nominative
Singular Plural
1st singular idealim ideallerim
2nd singular idealin ideallerin
3rd singular ideali idealleri
1st plural idealimiz ideallerimiz
2nd plural idealiniz idealleriniz
3rd plural idealleri idealleri
Definite accusative
Singular Plural
1st singular idealimi ideallerimi
2nd singular idealini ideallerini
3rd singular idealini ideallerini
1st plural idealimizi ideallerimizi
2nd plural idealinizi ideallerinizi
3rd plural ideallerini ideallerini
Dative
Singular Plural
1st singular idealime ideallerime
2nd singular idealine ideallerine
3rd singular idealine ideallerine
1st plural idealimize ideallerimize
2nd plural idealinize ideallerinize
3rd plural ideallerine ideallerine
Locative
Singular Plural
1st singular idealimde ideallerimde
2nd singular idealinde ideallerinde
3rd singular idealinde ideallerinde
1st plural idealimizde ideallerimizde
2nd plural idealinizde ideallerinizde
3rd plural ideallerinde ideallerinde
Ablative
Singular Plural
1st singular idealimden ideallerimden
2nd singular idealinden ideallerinden
3rd singular idealinden ideallerinden
1st plural idealimizden ideallerimizden
2nd plural idealinizden ideallerinizden
3rd plural ideallerinden ideallerinden
Genitive
Singular Plural
1st singular idealimin ideallerimin
2nd singular idealinin ideallerinin
3rd singular idealinin ideallerinin
1st plural idealimizin ideallerimizin
2nd plural idealinizin ideallerinizin
3rd plural ideallerinin ideallerinin
Predicative forms
Singular Plural
1st singular idealim ideallerim
2nd singular idealsin ideallersin
3rd singular ideal
idealdir
idealler
ideallerdir
1st plural idealiz idealleriz
2nd plural idealsiniz ideallersiniz
3rd plural idealler ideallerdir

SynonymsEdit