different between ideal vs exact
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)
- Optimal; being the best possibility.
- 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; […] .
- 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,
- Pertaining to ideas, or to a given idea.
- 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.
- 1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:
- Teaching or relating to the doctrine of idealism.
- (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)
- 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
- (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.
- (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.
- 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,
- (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 .
- (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 be an arbitrary semisimple Lie algebra. Then can be written uniquely as a direct sum of simple ideals (Theorem 5.2).
- 1975, Che-Young Lee (translator), Zhe-Xian Wan, Lie Algebras, Pergamon Press, page 13,
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)
- ideal
Noun
ideal m (plural ideales)
- ideal
Catalan
Etymology
From Latin ide?lis.
Pronunciation
- (Balearic, Central, Valencian) IPA(key): /i.de?al/
Adjective
ideal (masculine and feminine plural ideals)
- ideal
Derived terms
- idealment
Noun
ideal m (plural ideals)
- ideal
Galician
Etymology
From Latin ide?lis.
Pronunciation
Adjective
ideal m or f (plural ideais)
- ideal
Derived terms
- idealmente
Noun
ideal m (plural ideais)
- 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)
- 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
- ideal:
- optimal; being the best possibility.
- 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)
- (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)
- 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)
- 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)
- ideal
References
- “ideal” in The Nynorsk Dictionary.
Piedmontese
Pronunciation
- IPA(key): /ide?al/
Adjective
ideal
- ideal
Noun
ideal m
- 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)
- ideal
- notional
Derived terms
- idealmente
Noun
ideal m (plural ideais)
- ideal
- 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)
- 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 ??????)
- ideal
Declension
Spanish
Etymology
From Latin ide?lis.
Pronunciation
- IPA(key): /ide?al/, [i.ð?e?al]
Adjective
ideal (plural ideales)
- ideal
Derived terms
Noun
ideal m (plural ideales)
- ideal
Swedish
Pronunciation
Noun
ideal n
- ideal; perfect standard
- (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)
- ideal
Noun
ideal (definite accusative ideali, plural idealler)
- ideal
Declension
Synonyms
- ülkü
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exact
English
Etymology
From Old French, from Medieval Latin exactare, from Latin exactus, perfect passive participle of exig? (“demand, claim as due" or "measure by a standard, weigh, test”), from ex (“out”) + ag? (“drive”).
Pronunciation
- IPA(key): /???zækt/
- Rhymes: -ækt
Adjective
exact (comparative exacter or more exact, superlative exactest or most exact)
- Precisely agreeing with a standard, a fact, or the truth; perfectly conforming; neither exceeding nor falling short in any respect.
- Habitually careful to agree with a standard, a rule, or a promise; accurate; methodical; punctual.
- 1661, John Fell, The Life of the most learned, reverend and pious Dr. H. Hammond
- During the whole time of his abode in the university he generally spent thirteen hours of the day in study; by which assiduity besides an exact dispatch of the whole course of philosophy, he read over in a manner all classic authors that are extant […]
- 1661, John Fell, The Life of the most learned, reverend and pious Dr. H. Hammond
- Precisely or definitely conceived or stated; strict.
- (algebra, of a sequence of groups connected by homomorphisms) Such that the kernel of one homomorphism is the image of the preceding one.
Synonyms
- (precisely agreeing): perfect, true, correct, precise
- (precisely or definitely conceived or stated): strict
- spot on
Antonyms
- (precisely agreeing): inexact, imprecise, approximate
- (precisely or definitely conceived or stated): loose
Derived terms
- exactly
- exactness
- exact sequence
Translations
Verb
exact (third-person singular simple present exacts, present participle exacting, simple past and past participle exacted)
- (transitive, intransitive) To demand and enforce the payment or performance of, sometimes in a forcible or imperious way.
- to exact tribute, fees, or obedience from someone
- He said into them, Exact no more than that which is appointed you.
- 2018, Edo Konrad, "Living in the constant shadow of settler violence", +972 Magazine:
- 2020 September 19, statement of Clarence Thomas on the death of Ruth Bader Ginsburg[1]:
- She was a superb judge who gave her best and exacted the best from each of us, whether in agreement or disagreement.
- 2020, Kristine Henriksen Garroway, John W. Martens, Children and Methods (page 139)
- […] a generic, strikingly universal, deity, “ha-elohim,” who tests, who exacts and extracts, and who is the object of fear […]
- (transitive) To make desirable or necessary.
- 1632, Philip Massinger, The Maid of Honour
- My designs exact me in another place.
- 1632, Philip Massinger, The Maid of Honour
- (transitive) To inflict; to forcibly obtain or produce.
- to exact revenge on someone
Usage notes
- Often used with the preposition from or on.
Derived terms
- exactable
- exacter
- exacting
- exactor
Translations
Adverb
exact (comparative more exact, superlative most exact)
- exactly
Synonyms
- (error-free manner): accurately, just, precisely; see also Thesaurus:exactly
Further reading
- exact in Webster’s Revised Unabridged Dictionary, G. & C. Merriam, 1913.
- exact in The Century Dictionary, New York, N.Y.: The Century Co., 1911.
- exact at OneLook Dictionary Search
Dutch
Etymology
Borrowed from French exact, from Middle French exact, from Latin ex?ctus.
Pronunciation
- IPA(key): /?k?s?kt/
- Hyphenation: exact
- Rhymes: -?kt
Adjective
exact (comparative exacter, superlative exactst)
- exact, precise
Inflection
Derived terms
- exactheid
Descendants
- Afrikaans: eksak
French
Etymology
From Latin ex?ctus.
Pronunciation
- IPA(key): /??.zakt/, (old-fashioned) /??.za/
Adjective
exact (feminine singular exacte, masculine plural exacts, feminine plural exactes)
- exact; precise
Derived terms
- exactement
- science exacte
Further reading
- “exact” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).
Romanian
Etymology
Borrowed from French exact.
Pronunciation
- IPA(key): /e??zakt/
Adjective
exact m or n (feminine singular exact?, masculine plural exac?i, feminine and neuter plural exacte)
- exact, precise
Declension
Adverb
exact
- exactly, precisely
See also
- precis
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