ideal
English[edit]
Etymology[edit]
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[edit]
Adjective[edit]
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], “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:
- 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.
- 1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:
- 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[edit]
- (optimal): best, ideal, optimal, optimum
- (flawless): see also Thesaurus:flawless
- (of ideas): conceptual, notional
- (existing only in mind): conceptual, imaginary
Derived terms[edit]
Related terms[edit]
Translations[edit]
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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.
Noun[edit]
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).[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.
- 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) 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).
- 1975, Che-Young Lee (translator), Zhe-Xian Wan, Lie Algebras, Pergamon Press, page 13,
- (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[edit]
- (order theory): order ideal
- (type of Lie subalgebra): Lie ideal
Antonyms[edit]
- (order theory): filter
Hyponyms[edit]
- (mathematics): maximal ideal, principal ideal
Derived terms[edit]
Translations[edit]
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References[edit]
- ^
Boolean prime ideal theorem#Prime ideal theorems on Wikipedia.Wikipedia
- ^ Vaughan Pratt (2004) Chapter 1 : Lattice Theory[1], boole.stanford.edu, §1.3.5
Further reading[edit]
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[edit]
Asturian[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal (epicene, plural ideales)
Noun[edit]
ideal m (plural ideales)
Catalan[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal (masculine and feminine plural ideals)
Derived terms[edit]
Noun[edit]
ideal m (plural ideals)
Galician[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal m or f (plural ideais)
Derived terms[edit]
Noun[edit]
ideal m (plural ideais)
German[edit]
Etymology[edit]
Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.
Pronunciation[edit]
Adjective[edit]
ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten)
- ideal (optimal, perfect)
Declension[edit]
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 |
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 |
Derived terms[edit]
Further reading[edit]
Indonesian[edit]
Etymology[edit]
From Dutch ideaal, from French idéal, from Medieval Latin ideālis.
Pronunciation[edit]
Adjective[edit]
ideal
- ideal:
- optimal; being the best possibility.
- pertaining to ideas, or to a given idea.
Noun[edit]
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[edit]
Affixed terms[edit]
Related terms[edit]
Further reading[edit]
- “ideal” in Kamus Besar Bahasa Indonesia, Jakarta: Language Development and Fostering Agency — Ministry of Education, Culture, Research, and Technology of the Republic Indonesia, 2016.
Luxembourgish[edit]
Pronunciation[edit]
Adjective[edit]
ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)
Declension[edit]
number and gender | singular | plural | |||
---|---|---|---|---|---|
masculine | feminine | neuter | all genders | ||
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[edit]
Etymology[edit]
From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).
Noun[edit]
ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene)
References[edit]
- “ideal” in The Bokmål Dictionary.
Norwegian Nynorsk[edit]
Etymology[edit]
From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).
Noun[edit]
ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)
References[edit]
- “ideal” in The Nynorsk Dictionary.
Piedmontese[edit]
Pronunciation[edit]
Adjective[edit]
ideal
Noun[edit]
ideal m
Portuguese[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal m or f (plural ideais)
Derived terms[edit]
Noun[edit]
ideal m (plural ideais)
Romanian[edit]
Etymology[edit]
From French idéal, from Latin idealis.
Adjective[edit]
ideal m or n (feminine singular ideală, masculine plural ideali, feminine and neuter plural ideale)
Declension[edit]
Related terms[edit]
Serbo-Croatian[edit]
Pronunciation[edit]
Noun[edit]
idèāl m (Cyrillic spelling идѐа̄л)
Declension[edit]
Spanish[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal (plural ideales)
Derived terms[edit]
Noun[edit]
ideal m (plural ideales)
Further reading[edit]
- “ideal”, in Diccionario de la lengua española, Vigésima tercera edición, Real Academia Española, 2014
Swedish[edit]
Pronunciation[edit]
audio (file)
Noun[edit]
ideal n
- ideal; perfect standard
- (mathematics) ideal; special subsets of a ring
Declension[edit]
Declension of ideal | ||||
---|---|---|---|---|
Singular | Plural | |||
Indefinite | Definite | Indefinite | Definite | |
Nominative | ideal | idealet | ideal | idealen |
Genitive | ideals | idealets | ideals | idealens |
Anagrams[edit]
Tagalog[edit]
Alternative forms[edit]
Etymology[edit]
Pronunciation[edit]
Noun[edit]
ideál
Related terms[edit]
Adjective[edit]
ideál
Turkish[edit]
Etymology[edit]
Pronunciation[edit]
Adjective[edit]
ideal
Noun[edit]
ideal (definite accusative ideali, plural idealler)
Declension[edit]
Synonyms[edit]
- English terms derived from French
- English terms derived from Late Latin
- English terms derived from Latin
- Rhymes:English/iːəl
- Rhymes:English/iːəl/3 syllables
- English 2-syllable words
- English 3-syllable words
- English terms with IPA pronunciation
- English terms with audio links
- English lemmas
- English adjectives
- English terms with quotations
- English terms with usage examples
- en:Mathematics
- English nouns
- English countable nouns
- en:Algebra
- en:Set theory
- en:Infinity
- en:Non-Euclidean geometry
- Asturian terms derived from Latin
- Asturian terms with IPA pronunciation
- Asturian lemmas
- Asturian adjectives
- Asturian nouns
- Asturian masculine nouns
- Catalan terms derived from Latin
- Catalan 3-syllable words
- Catalan terms with IPA pronunciation
- Catalan lemmas
- Catalan adjectives
- Catalan epicene adjectives
- Catalan nouns
- Catalan masculine nouns
- Galician terms derived from Latin
- Galician lemmas
- Galician adjectives
- Galician nouns
- Galician masculine nouns
- German terms borrowed from Late Latin
- German terms derived from Late Latin
- German terms derived from Latin
- German doublets
- German 3-syllable words
- German terms with IPA pronunciation
- Rhymes:German/aːl
- Rhymes:German/aːl/3 syllables
- German terms with audio links
- German lemmas
- German adjectives
- Indonesian terms borrowed from Dutch
- Indonesian terms derived from Dutch
- Indonesian terms derived from French
- Indonesian terms derived from Medieval Latin
- Indonesian terms with IPA pronunciation
- Indonesian lemmas
- Indonesian adjectives
- Indonesian nouns
- id:Mathematics
- Luxembourgish 3-syllable words
- Luxembourgish terms with IPA pronunciation
- Luxembourgish lemmas
- Luxembourgish adjectives
- Norwegian Bokmål terms derived from French
- Norwegian Bokmål terms derived from Late Latin
- Norwegian Bokmål terms derived from Latin
- Norwegian Bokmål lemmas
- Norwegian Bokmål nouns
- Norwegian Bokmål neuter nouns
- Norwegian Nynorsk terms derived from French
- Norwegian Nynorsk terms derived from Late Latin
- Norwegian Nynorsk terms derived from Latin
- Norwegian Nynorsk lemmas
- Norwegian Nynorsk nouns
- Norwegian Nynorsk neuter nouns
- Piedmontese terms with IPA pronunciation
- Piedmontese lemmas
- Piedmontese adjectives
- Piedmontese nouns
- Piedmontese masculine nouns
- Portuguese terms borrowed from Latin
- Portuguese terms derived from Latin
- Portuguese terms with IPA pronunciation
- Rhymes:Portuguese/al
- Rhymes:Portuguese/al/3 syllables
- Rhymes:Portuguese/aw
- Rhymes:Portuguese/aw/3 syllables
- Portuguese 3-syllable words
- Portuguese lemmas
- Portuguese adjectives
- Portuguese nouns
- Portuguese countable nouns
- Portuguese masculine nouns
- Romanian terms borrowed from French
- Romanian terms derived from French
- Romanian terms derived from Latin
- Romanian lemmas
- Romanian adjectives
- Serbo-Croatian terms with IPA pronunciation
- Serbo-Croatian lemmas
- Serbo-Croatian nouns
- Serbo-Croatian masculine nouns
- Spanish terms derived from Latin
- Spanish 3-syllable words
- Spanish terms with IPA pronunciation
- Rhymes:Spanish/al
- Rhymes:Spanish/al/3 syllables
- Spanish lemmas
- Spanish adjectives
- Spanish nouns
- Spanish countable nouns
- Spanish masculine nouns
- Swedish terms with audio links
- Swedish lemmas
- Swedish nouns
- Swedish neuter nouns
- sv:Mathematics
- Tagalog terms borrowed from Spanish
- Tagalog terms derived from Spanish
- Tagalog terms with IPA pronunciation
- Tagalog lemmas
- Tagalog nouns
- Tagalog adjectives
- Turkish terms borrowed from French
- Turkish terms derived from French
- Turkish terms with IPA pronunciation
- Turkish lemmas
- Turkish adjectives
- Turkish nouns