Wiktionary

ideal

See also: Ideal, ideał, ideál, and idéal

English

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English Wikipedia has an article on:
Wikipedia

Etymology

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From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea +‎ -al, 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.

Pronunciation

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

Adjective

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ideal (comparative more ideal, superlative most ideal)

  1. Pertaining to ideas, or to a given idea.
  2. Existing only in the mind; conceptual, imaginary.
    • 1796, Matthew Lewis, The Monk, Folio Society, published 1985, page 256:
      The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
    • 1816 June – 1817 April/May (date written), [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, published 1 January 1818, →OCLC:
      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:
      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.
  3. Optimal; being the best possibility.
  4. 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; […] .
  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.

Synonyms

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Derived terms

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Translations

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

Noun

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ideal (plural ideals)

  1. A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.
  3. 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
    • 1945 April 16, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S.‎[3], Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162:
      With great humility, I call upon all Americans to help me keep our nation united in defense of those ideals which have been so eloquently proclaimed by Franklin Roosevelt. I want in turn to assure my fellow Americans and all of those who love peace and liberty throughout the world that I will support and defend those ideals with all my strength and all my heart.
  4. (algebra, ring theory) A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the 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”, in 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.
  5. (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[4], 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.
  6. (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  .
  7. (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, Zhe-Xian Wan, translated by Che-Young Lee, 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”, in 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).
  8. (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.

Synonyms

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Antonyms

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  • (antonym(s) of “order theory”): filter

Hyponyms

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Derived terms

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Translations

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References

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

Further reading

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Anagrams

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Asturian

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Etymology

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From Latin ideālis.

Pronunciation

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

Adjective

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ideal (epicene, plural ideales)

  1. ideal

Noun

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ideal m (plural ideales)

  1. ideal

Further reading

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Catalan

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Etymology

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Borrowed from Latin ideālis.

Pronunciation

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Adjective

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ideal m or f (masculine and feminine plural ideals)

  1. ideal

Derived terms

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Noun

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ideal m (plural ideals)

  1. ideal

Galician

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Etymology

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From Latin ideālis.

Pronunciation

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  • IPA(key): /idɛˈal/ [i.ð̞ɛˈɑɫ]
  • Rhymes: -al
  • Hyphenation: i‧de‧al

Adjective

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ideal m or f (plural ideais)

  1. ideal

Derived terms

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Noun

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ideal m (plural ideais)

  1. ideal

Further reading

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German

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Etymology

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

Pronunciation

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Adjective

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

  1. ideal (optimal, perfect)

Declension

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Positive forms of ideal
number & gender singular plural
masculine feminine neuter
predicative er ist ideal sie ist ideal es ist ideal sie sind ideal
strong declension
(without article) 		nominative idealer ideale ideales ideale
genitive idealen idealer idealen idealer
dative idealem idealer idealem idealen
accusative idealen ideale ideales ideale
weak declension
(with definite article) 		nominative der ideale die ideale das ideale die idealen
genitive des idealen der idealen des idealen der idealen
dative dem idealen der idealen dem idealen den idealen
accusative den idealen die ideale das ideale die idealen
mixed declension
(with indefinite article) 		nominative ein idealer eine ideale ein ideales (keine) idealen
genitive eines idealen einer idealen eines idealen (keiner) idealen
dative einem idealen einer idealen einem idealen (keinen) idealen
accusative einen idealen eine ideale ein ideales (keine) idealen
Comparative forms of ideal
number & gender singular plural
masculine feminine neuter
predicative er ist idealer sie ist idealer es ist idealer sie sind idealer
strong declension
(without article) 		nominative idealerer idealere idealeres idealere
genitive idealeren idealerer idealeren idealerer
dative idealerem idealerer idealerem idealeren
accusative idealeren idealere idealeres idealere
weak declension
(with definite article) 		nominative der idealere die idealere das idealere die idealeren
genitive des idealeren der idealeren des idealeren der idealeren
dative dem idealeren der idealeren dem idealeren den idealeren
accusative den idealeren die idealere das idealere die idealeren
mixed declension
(with indefinite article) 		nominative ein idealerer eine idealere ein idealeres (keine) idealeren
genitive eines idealeren einer idealeren eines idealeren (keiner) idealeren
dative einem idealeren einer idealeren einem idealeren (keinen) idealeren
accusative einen idealeren eine idealere ein idealeres (keine) idealeren
Superlative forms of ideal
number & gender singular plural
masculine feminine neuter
predicative er ist am idealsten sie ist am idealsten es ist am idealsten sie sind am idealsten
strong declension
(without article) 		nominative idealster idealste idealstes idealste
genitive idealsten idealster idealsten idealster
dative idealstem idealster idealstem idealsten
accusative idealsten idealste idealstes idealste
weak declension
(with definite article) 		nominative der idealste die idealste das idealste die idealsten
genitive des idealsten der idealsten des idealsten der idealsten
dative dem idealsten der idealsten dem idealsten den idealsten
accusative den idealsten die idealste das idealste die idealsten
mixed declension
(with indefinite article) 		nominative ein idealster eine idealste ein idealstes (keine) idealsten
genitive eines idealsten einer idealsten eines idealsten (keiner) idealsten
dative einem idealsten einer idealsten einem idealsten (keinen) idealsten
accusative einen idealsten eine idealste ein idealstes (keine) idealsten

Derived terms

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Further reading

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

Indonesian

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Indonesian Wikipedia has an article on:
Wikipedia id

Etymology

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

Pronunciation

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Adjective

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idéal

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

Noun

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idéal (plural ideal-ideal)

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

Alternative forms

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Derived terms

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Further reading

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Luxembourgish

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Pronunciation

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

Adjective

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

  1. ideal

Declension

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Declension of ideal
singular plural
masculine feminine neuter
predicative hien ass ideal si ass ideal et ass ideal si si(nn) ideal
nominative /
accusative 		attributive and/or after determiner idealen ideal ideaalt ideal
independent without determiner ideales idealer
dative after any declined word idealen idealer idealen idealen
as first declined word idealem idealem

Norwegian Bokmål

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Etymology

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

Noun

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

  1. ideal

References

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Norwegian Nynorsk

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Etymology

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

Noun

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

  1. ideal

References

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Piedmontese

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Pronunciation

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Adjective

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ideal

  1. ideal

Noun

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ideal m

  1. ideal

Portuguese

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Etymology

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Borrowed from Latin ideālis.

Pronunciation

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

Adjective

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ideal m or f (plural ideais)

  1. ideal
  2. notional

Derived terms

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Noun

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ideal m (plural ideais)

  1. ideal
  2. fantasy

Romanian

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Etymology

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Borrowed from French idéal, from Latin idealis.

Pronunciation

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Adjective

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

  1. ideal

Declension

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Declension of ideal
singular plural
masculine neuter feminine masculine neuter feminine
nominative-
accusative 		indefinite ideal ideală ideali ideale
definite idealul ideala idealii idealele
genitive-
dative 		indefinite ideal ideale ideali ideale
definite idealului idealei idealilor idealelor
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Serbo-Croatian

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Pronunciation

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

Noun

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idèāl m (Cyrillic spelling идѐа̄л)

  1. ideal

Declension

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Declension of ideal
singular plural
nominative idèāl ideali
genitive ideála ideala
dative idealu idealima
accusative ideal ideale
vocative ideale ideali
locative idealu idealima
instrumental idealom idealima

Spanish

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Etymology

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From Latin ideālis.

Pronunciation

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

Adjective

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ideal m or f (masculine and feminine plural ideales)

  1. ideal

Derived terms

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Noun

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ideal m (plural ideales)

  1. ideal

Further reading

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Swedish

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Pronunciation

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Adjective

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ideal (not comparable)

  1. ideal (perfect)
    Synonym: idealisk
    under ideala förhållanden
    under ideal conditions

Declension

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Inflection of ideal
Indefinite positive comparative superlative1
common singular ideal — —
neuter singular idealt — —
plural ideala — —
masculine plural2 ideale — —
Definite positive comparative superlative
masculine singular3 ideale — —
all ideala — —

1 The indefinite superlative forms are only used in the predicative.
2 Dated or archaic.
3 Only used, optionally, to refer to things whose natural gender is masculine.

Noun

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ideal n

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

Declension

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Declension of ideal
nominative genitive
singular indefinite ideal ideals
definite idealet idealets
plural indefinite ideal ideals
definite idealen idealens
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References

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Anagrams

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Tagalog

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Alternative forms

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Etymology

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Borrowed from Spanish ideal.

Pronunciation

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Noun

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ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

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

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ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

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

Turkish

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Etymology

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Borrowed from French idéal.

Pronunciation

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

Adjective

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ideal

  1. ideal

Noun

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ideal (definite accusative ideali, plural idealler)

  1. ideal

Declension

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Declension of ideal
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

Synonyms

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Welsh

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Alternative forms

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Etymology

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Borrowed from English ideal.[1]

Pronunciation

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Adjective

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ideal (feminine singular ideal, plural ideal, not comparable)

  1. ideal[1][2]
    Synonym: delfrydol

Noun

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ideal m (plural idealau)

  1. ideal[1][2]
    Synonym: delfryd

Mutation

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Mutated forms of ideal
radical soft nasal h-prothesis
ideal unchanged unchanged hideal

Note: Certain mutated forms of some words can never occur in standard Welsh.
All possible mutated forms are displayed for convenience.

References

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  1. ↑ 1.0 1.1 1.2 R. J. Thomas, G. A. Bevan, P. J. Donovan, A. Hawke et al., editors (1950–present), “ideal”, in Geiriadur Prifysgol Cymru Online (in Welsh), University of Wales Centre for Advanced Welsh & Celtic Studies
  2. ↑ 2.0 2.1 Delyth Prys, J.P.M. Jones, Owain Davies, Gruffudd Prys (2006) Y Termiadur: termau wedi'u safoni; standardised terminology‎[2] (in Welsh), Cardiff: Awdurdod cymwysterau, cwricwlwm ac asesu Cymru (Qualifications curriculum & assessment authority for Wales), →ISBN
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