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, 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: […] [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.
- 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
- 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.[2], 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).[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[3], 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, 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.[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 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 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 m or f (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
Borrowed from French idéal, from Latin idealis.
Pronunciation edit
Audio (file)
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 m or f (masculine and feminine 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)
Adjective edit
ideal (not comparable)
Declension edit
Inflection of ideal | |||
---|---|---|---|
Indefinite | Positive | Comparative | Superlative2 |
Common singular | ideal | — | — |
Neuter singular | idealt | — | — |
Plural | ideala | — | — |
Masculine plural3 | ideale | — | — |
Definite | Positive | Comparative | Superlative |
Masculine singular1 | ideale | — | — |
All | ideala | — | — |
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 |
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 |
Related terms edit
References edit
- ideal in Svensk ordbok (SO)
- ideal in Svenska Akademiens ordlista (SAOL)
- ideal in Svenska Akademiens ordbok (SAOB)
Anagrams edit
Tagalog edit
Alternative forms edit
Etymology edit
Pronunciation edit
Noun edit
ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
Related terms edit
Adjective edit
ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
Turkish edit
Etymology edit
Pronunciation edit
Adjective edit
ideal
Noun edit
ideal (definite accusative ideali, plural idealler)