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English
Etymology
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.
Pronunciation
- Rhymes: -iːəl
- IPA(key): /aɪˈdiːl/, /aɪˈdɪəl/, /aɪˈdiː.əl/
Adjective
ideal (comparative more ideal, superlative most ideal)
- Pertaining to ideas, or to a given idea.
- 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 —
1818, [Mary Shelley], “Chapter 4”, in Frankenstein; or, The Modern Prometheus. , volumes (please specify |volume=I to III), London: for Lackington, Hughes, Harding, Mavor, & Jones, →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.
- 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; .
- Teaching or relating to the doctrine of idealism.
the ideal theory or philosophy
- (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
Derived terms
Related terms
Translations
being optimal
- Afrikaans: voorbeeldig, ideaal
- Arabic: مِثَالِيّ (miṯāliyy)
- Asturian: ideal
- Belarusian: ідэа́льны (ideálʹny)
- Bulgarian: идеа́лен (bg) (ideálen)
- Catalan: ideal (ca)
- Chinese:
- Mandarin: 理想的 (zh) (lǐxiǎng de)
- Czech: ideální (cs)
- Dutch: ideaal (nl), optimaal (nl)
- Esperanto: ideala
- Finnish: ihanteellinen (fi), ideaalinen (fi), ideaali- (fi), optimaalinen (fi)
- French: idéal (fr)
- Galician: ideal
- German: ideal (de), bestmöglich, optimal (de)
- Hindi: आदर्श (hi) (ādarś), श्रेष्ठ (hi) (śreṣṭh), आइड़ियल (hi) (āiṛiyal)
- Hungarian: ideális (hu)
- Ido: ideala (io)
- Irish: idéalach
- Italian: ideale (it)
- Japanese: 理想的な (ja) (りそうてきな, risōteki na), 理想な (ja) (りそうな, risō na)
- Korean: 이상적(理想的) (ko) (isangjeok), 리상적(理想的) (risangjeok) (North Korea)
- Latvian: ideāls
- Luxembourgish: ideal
- Macedonian: идеа́лен (ideálen)
- Malayalam: ആദർശം (ml) (ādaṟśaṁ)
- Norman: idéal
- Persian: آرمانی (fa) (ârmâni), ایدهآل (ide-âl)
- Polish: idealny (pl) m
- Portuguese: ideal (pt)
- Russian: идеа́льный (ru) (ideálʹnyj)
- Slovak: ideálny
- Slovene: idealen
- Spanish: ideal (es)
- Telugu: ఆదర్శ (te) (ādarśa)
- Ukrainian: ідеа́льний (ideálʹnyj)
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being perfect
- Afrikaans: ideaal
- Asturian: ideal
- Belarusian: ідэа́льны (ideálʹny), даскана́лы (daskanály)
- Bulgarian: съвърше́н (bg) (sǎvǎršén), идеа́лен (bg) (ideálen)
- Catalan: ideal (ca)
- Czech: ideální (cs)
- Dutch: ideaal (nl), perfect (nl), volmaakt (nl)
- Esperanto: ideala
- Finnish: ihanteellinen (fi), ideaalinen (fi)
- French: idéal (fr), parfait (fr)
- Galician: ideal m or f
- German: ideal (de), vollendet (de), einwandfrei (de), tadellos (de), perfekt (de)
- Greek: ιδανικός (el) (idanikós)
- Hindi: आदर्श (hi) (ādarś), आइड़ियल (hi) (āiṛiyal), श्रेष्ठ (hi) (śreṣṭh)
- Hungarian: tökéletes (hu), eszményi (hu)
- Ido: ideala (io)
- Irish: idéalach
- Manx: slanjeant
- Maori: takarepakore, paruhi
- Norman: idéal
- Persian: کامل (fa) (kâmel), فربود (farbud), درست (fa) (dorost)
- Polish: idealny (pl) m
- Portuguese: ideal (pt)
- Russian: идеа́льный (ru) (ideálʹnyj), соверше́нный (ru) (soveršénnyj)
- Slovene: idealen
- Spanish: ideal (es)
- Telugu: ఆదర్శ (te) (ādarśa)
- Ukrainian: ідеа́льний (ideálʹnyj), доскона́лий (doskonályj)
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Translations to be checked
Noun
ideal (plural ideals)
- A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.
- 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., 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.
- (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”, 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.
- (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 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.
- (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) 𝖍 of a given Lie algebra 𝖌 such that the Lie bracket 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).
- (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.
- 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
Antonyms
- (antonym(s) of "order theory"): filter
Hyponyms
Derived terms
Translations
perfect standard of beauty, intellect etc.
- Afrikaans: ideaal
- Asturian: ideal m
- Belarusian: ідэа́л m (ideál)
- Bulgarian: идеа́л (bg) m (ideál)
- Catalan: ideal (ca) m
- Chinese:
- Mandarin: 典范 (zh) (diǎnfàn), 理想 (zh) (lǐxiǎng)
- Czech: ideál m
- Dutch: ideaal (nl) n, perfectie (nl) f, streefdoel (nl) n
- Esperanto: ideala, perfekta
- Finnish: ihanne (fi), ideaali (fi)
- French: idéal (fr) m
- Galician: ideal m
- German: Ideal (de) n
- Greek: ιδεώδες (el) n (ideódes)
- Hebrew: אידיאל \ אִידֵאָל (he) (ide'ál)
- Hindi: आइड़ियल (hi) m (āiṛiyal), आदर्श (hi) m (ādarś), श्रेष्ठ (hi) m (śreṣṭh)
- Hungarian: eszmény (hu), ideál (hu), mintakép (hu)
- Ido: idealo (io)
- Indonesian: ideal (id)
- Interlingua: ideal
- Irish: idéal m
- Italian: ideale (it) m
- Japanese: 理想 (ja) (りそう, risō)
- Kazakh: мұрат (mūrat)
- Korean: 이상(理想) (ko) (isang), 리상(理想) (ko) (risang) (North Korea)
- Persian: آرمان (fa) (ârmân), ایدئال (ide'âl)
- Polish: ideał (pl) m
- Portuguese: ideal (pt) m
- Punjabi: ਆਦਰਸ਼ (pa) m (ādaraś)
- Russian: идеа́л (ru) m (ideál)
- Slovak: ideál m
- Slovene: ideal m
- Spanish: ideal (es) m
- Swedish: ideal (sv) n
- Telugu: ఆదర్శం (te) (ādarśaṁ)
- Ukrainian: ідеа́л m (ideál)
- Vietnamese: lý tưởng (vi)
- Welsh: delfryd (cy) m or f
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subring closed under multiplication by containing ring
lower set closed under joins
References
Further reading
Anagrams
Asturian
Etymology
From Latin ideālis.
Pronunciation
Adjective
ideal (epicene, plural ideales)
- ideal
Noun
ideal m (plural ideales)
- ideal
Catalan
Etymology
Borrowed from Latin ideālis.
Pronunciation
Adjective
ideal m or f (masculine and feminine plural ideals)
- ideal
Derived terms
Noun
ideal m (plural ideals)
- ideal
Galician
Etymology
From Latin ideālis.
Pronunciation
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This entry needs pronunciation information. If you are familiar with the IPA then please add some!
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Adjective
ideal m or f (plural ideais)
- ideal
Derived terms
Noun
ideal m (plural ideais)
- ideal
German
Etymology
Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.
Pronunciation
Adjective
ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten)
- ideal (optimal, perfect)
Declension
Comparative forms of ideal
Superlative forms of ideal
Derived terms
Further reading
- “ideal” in Duden online
- “ideal” in Digitales Wörterbuch der deutschen Sprache
Indonesian
Etymology
From Dutch ideaal, from French idéal, from Medieval Latin ideālis.
Pronunciation
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
Affixed terms
Related terms
Further reading
Luxembourgish
Pronunciation
- IPA(key): /ideˈaːl/, , /idiˈaːl/
Adjective
ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)
- ideal
Declension
Norwegian Bokmål
Etymology
From French idéal, from Late Latin ideālis (“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
Norwegian Nynorsk
Etymology
From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).
Noun
ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)
- ideal
References
Piedmontese
Pronunciation
Adjective
ideal
- ideal
Noun
ideal m
- ideal
Portuguese
Etymology
Borrowed from Latin ideālis.
Pronunciation
- Rhymes: -al, -aw
- Hyphenation: i‧de‧al
Adjective
ideal m or f (plural ideais)
- ideal
- notional
Derived terms
Noun
ideal m (plural ideais)
- ideal
- fantasy
Romanian
Etymology
Borrowed from French idéal, from Latin idealis.
Pronunciation
Adjective
ideal m or n (feminine singular ideală, masculine plural ideali, feminine and neuter plural ideale)
- ideal
Declension
Related terms
Serbo-Croatian
Pronunciation
- IPA(key): /iděaːl/
- Hyphenation: i‧de‧al
Noun
idèāl m (Cyrillic spelling идѐа̄л)
- ideal
Declension
Spanish
Etymology
From Latin ideālis.
Pronunciation
- IPA(key): /ideˈal/
- Rhymes: -al
- Syllabification: i‧de‧al
Adjective
ideal m or f (masculine and feminine plural ideales)
- ideal
Derived terms
Noun
ideal m (plural ideales)
- ideal
Further reading
Swedish
Pronunciation
Adjective
ideal (not comparable)
- ideal (perfect)
- Synonym: idealisk
under ideala förhållanden- under ideal conditions
Declension
Inflection of ideal
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Indefinite
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Positive
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Comparative
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Superlative2
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Common singular
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ideal
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Neuter singular
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idealt
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Plural
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ideala
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Masculine plural3
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ideale
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Definite
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Positive
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Comparative
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Superlative
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Masculine singular1
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ideale
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All
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ideala
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1) Only used, optionally, to refer to things whose natural gender is masculine. 2) The indefinite superlative forms are only used in the predicative. 3) Dated or archaic
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Noun
ideal n
- ideal; perfect standard
- (mathematics) ideal; special subsets of a ring
Declension
Related terms
References
Anagrams
Tagalog
Alternative forms
Etymology
Borrowed from Spanish ideal.
Pronunciation
- IPA(key): /ʔideˈal/,
- Hyphenation: i‧de‧al
Noun
ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
- ideal (model to be imitated)
- Synonyms: huwaran, uliran
- ambition
- Synonyms: mithi, mithiin, pangarap, ambisyon
Related terms
Adjective
ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
- ideal (just as what one wants)
- worthy of imitation
- Synonym: uliran
Turkish
Etymology
Borrowed from French idéal.
Pronunciation
- IPA(key): /ideˈɑl/
- Hyphenation: i‧de‧al
Adjective
ideal
- ideal
Noun
ideal (definite accusative ideali, plural idealler)
- ideal
Declension
Synonyms