different between ideal vs easy

ideal

English

Etymology

From French idéal, from Late Latin ide?lis (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.

Pronunciation

  • Rhymes: -i??l
  • IPA(key): /a??d??l/, /a??di?.?l/

Adjective

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, Frankenstein, or the Modern Prometheus,[1] Chapter 4,
      Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
  5. Teaching or relating to the doctrine of idealism.
  6. (mathematics) Not actually present, but considered as present when limits at infinity are included.

Synonyms

  • See also Thesaurus:flawless

Related terms

Derived terms

Translations

Noun

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 Z {\displaystyle \mathbb {Z} } be the ring of integers and let 2 Z {\displaystyle 2\mathbb {Z} } be its ideal of even integers. Then the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is a Boolean ring.
    The product of two ideals a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} is an ideal a b {\displaystyle {\mathfrak {ab}}} which is a subset of the intersection of a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} . This should help to understand why maximal ideals are prime ideals. Likewise, the union of a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} is a subset of a + b {\displaystyle {\mathfrak {a+b}}} .
    • 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).
    • 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 I {\displaystyle I} of a given set X {\displaystyle X} is a nonempty subset of the powerset P ( X ) {\displaystyle {\mathcal {P}}(X)} such that: ( 1 )   ? ? I {\displaystyle (1)\ \emptyset \in I} , ( 2 )   A ? I ? B ? A ? B ? I {\displaystyle (2)\ A\in I\land B\subseteq A\implies B\in I} and ( 3 )   A , B ? I ? A ? B ? I {\displaystyle (3)\ A,B\in I\implies A\cup B\in I} .
  5. (algebra, Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) ???? of a given Lie algebra ???? such that the Lie bracket [????,????] is a subset of ????.
    • 1975, Che-Young Lee (translator), Zhe-Xian Wan, Lie Algebras, Pergamon Press, page 13,
      If ???? is a Lie algebra, ???? is an ideal and the Lie algebras ???? and ????/???? are solvable, then ???? 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 L {\displaystyle L} be an arbitrary semisimple Lie algebra. Then L {\displaystyle L} can be written uniquely as a direct sum L 1 ? ? ? L t {\displaystyle L_{1}\oplus \dots \oplus L_{t}} of simple ideals (Theorem 5.2).

Synonyms

  • (type of Lie subalgebra): Lie ideal

Antonyms

  • (order theory): filter

Hyponyms

  • (mathematics): maximal ideal, principal ideal

Derived terms

Translations

References

Further reading

  • Ideal (ring theory) on Wikipedia.Wikipedia
  • Ideal (order theory) on Wikipedia.Wikipedia
  • Ideal (set theory) on Wikipedia.Wikipedia
  • Ideal point on Wikipedia.Wikipedia
  • Ideal triangle on Wikipedia.Wikipedia
  • Lie algebra#Subalgebras, ideals and homomorphisms on Wikipedia.Wikipedia

Anagrams

  • Delia, Elida, ailed, ladie

Asturian

Etymology

From Latin ide?lis.

Pronunciation

Adjective

ideal (epicene, plural ideales)

  1. ideal

Noun

ideal m (plural ideales)

  1. ideal

Catalan

Etymology

From Latin ide?lis.

Pronunciation

  • (Balearic, Central, Valencian) IPA(key): /i.de?al/

Adjective

ideal (masculine and feminine plural ideals)

  1. ideal

Derived terms

  • idealment

Noun

ideal m (plural ideals)

  1. ideal

Galician

Etymology

From Latin ide?lis.

Pronunciation

Adjective

ideal m or f (plural ideais)

  1. ideal

Derived terms

  • idealmente

Noun

ideal m (plural ideais)

  1. ideal

German

Etymology

Borrowed from Late Latin ide?lis (existing in idea), from Latin idea (idea). Doublet of ideell.

Pronunciation

  • IPA(key): /ide?a?l/
  • Rhymes: -a?l

Adjective

ideal (comparative idealer, superlative am idealsten)

  1. ideal (optimal, perfect)

Declension

Derived terms

  • idealerweise

Further reading

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

Indonesian

Etymology

From Dutch ideaal, from French idéal, from Medieval Latin ide?lis.

Pronunciation

  • IPA(key): [??d?eal]
  • Hyphenation: idé?al

Adjective

ideal

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

Noun

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 forms

  • idiil

Affixed terms

Related terms

Further reading

  • “ideal” in Kamus Besar Bahasa Indonesia (KBBI) Daring, Jakarta: Badan Pengembangan dan Pembinaan Bahasa, Kementerian Pendidikan dan Kebudayaan Republik Indonesia, 2016.

Luxembourgish

Pronunciation

  • IPA(key): /ide?a?l/, [id??a?l], /idi?a?l/

Adjective

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

  1. ideal

Declension


Norwegian Bokmål

Etymology

From French idéal, from Late Latin ide?lis (existing in idea), from Latin idea (idea)

Noun

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

  1. ideal

References

  • “ideal” in The Bokmål Dictionary.

Norwegian Nynorsk

Etymology

From French idéal, from Late Latin ide?lis (existing in idea), from Latin idea (idea)

Noun

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

  1. ideal

References

  • “ideal” in The Nynorsk Dictionary.

Piedmontese

Pronunciation

  • IPA(key): /ide?al/

Adjective

ideal

  1. ideal

Noun

ideal m

  1. ideal

Portuguese

Etymology

From Latin ide?lis.

Pronunciation

  • (Portugal) IPA(key): /i?ðja?/
  • (Brazil) IPA(key): /i.de?aw/
  • Hyphenation: i?de?al

Adjective

ideal m or f (plural ideais, comparable)

  1. ideal
  2. notional

Derived terms

  • idealmente

Noun

ideal m (plural ideais)

  1. ideal
  2. fantasy

Romanian

Etymology

From French idéal, from Latin idealis.

Adjective

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

  1. ideal

Declension

Related terms

  • idealism
  • idealist
  • idealiza

Serbo-Croatian

Pronunciation

  • IPA(key): /id?a?l/
  • Hyphenation: i?de?al

Noun

idè?l m (Cyrillic spelling ??????)

  1. ideal

Declension


Spanish

Etymology

From Latin ide?lis.

Pronunciation

  • IPA(key): /ide?al/, [i.ð?e?al]

Adjective

ideal (plural ideales)

  1. ideal

Derived terms

Noun

ideal m (plural ideales)

  1. ideal

Swedish

Pronunciation

Noun

ideal n

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

Declension

Anagrams

  • ilade

Turkish

Etymology

Borrowed from French idéal.

Pronunciation

  • IPA(key): /ide?a?/
  • Hyphenation: i?de?al

Adjective

ideal (comparative daha ideal, superlative en ideal)

  1. ideal

Noun

ideal (definite accusative ideali, plural idealler)

  1. ideal

Declension

Synonyms

  • ülkü

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easy

English

Alternative forms

  • aisy (dialectal, archaic)
  • easie (obsolete)
  • eazy (eye dialect)
  • EZ (abbreviation, US, informal)

Etymology

From Middle English eesy, esy, partly from Middle English ese (ease) + -y, equivalent to ease +? -y, and partly from Old French aisié (eased, at ease, at leisure), past participle of aisier (to put at ease), from aise (empty space, elbow room, opportunity), of uncertain origin. See ease. Merged with Middle English ethe, eathe (easy), from Old English ?eþe, from Proto-Germanic *auþuz, from Proto-Indo-European *aut- (empty, lonely). Compare also Old Saxon ?þi, Old High German ?di, Old Norse auðr, all meaning "easy, vacant, empty." More at ease, eath.

Pronunciation

  • (Received Pronunciation) IPA(key): /?i?zi/, /?i?z?/
  • (General American) IPA(key): /?izi/
  • Rhymes: -i?zi

Adjective

easy (comparative easier or more easy, superlative easiest or most easy)

  1. (now rare except in certain expressions) Comfortable; at ease.
  2. Requiring little skill or effort.
  3. Causing ease; giving comfort, or freedom from care or labour.
    Rich people live in easy circumstances.
    an easy chair
  4. Free from constraint, harshness, or formality; unconstrained; smooth.
    easy manners; an easy style
  5. (informal, derogatory, of a woman) Consenting readily to sex.
  6. Not making resistance or showing unwillingness; tractable; yielding; compliant.
    • He gain'd their easy hearts.
  7. (finance, dated) Not straitened as to money matters; opposed to tight.
    The market is easy.

Synonyms

  • (comfortable): relaxed, relaxing
  • (not difficult): light, eath
  • (consenting readily to sex): fast
  • (requiring little skill or effort): soft, trivial
  • See also Thesaurus:easy

Antonyms

  • (comfortable, at ease): uneasy, anxious
  • (requiring little skill or effort): difficult, hard, uneasy, uneath, challenging

Derived terms

Related terms

  • ease

Descendants

  • ? Faroese: isi
  • ? Finnish: iisi

Translations

Adverb

easy (comparative easier, superlative easiest)

  1. In a relaxed or casual manner.
  2. In a manner without strictness or harshness.
  3. Used an intensifier for large magnitudes.
  4. Not difficult, not hard. (Can we add an example for this sense?)

Derived terms

  • breathe easy

Noun

easy (plural easies)

  1. Something that is easy

Verb

easy (third-person singular simple present easies, present participle easying, simple past and past participle easied)

  1. (rowing) Synonym of easy-oar

Anagrams

  • Ayes, Saye, Seay, ayes, eyas, saye, yaes, yeas

Middle English

Adjective

easy

  1. Alternative form of esy

Adverb

easy

  1. Alternative form of esy

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