English Online Dictionary. What means ideal? What does ideal mean?
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)}: /aɪˈdɪəl/, /aɪˈdiː.əl/
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.
- Teaching or relating to the doctrine of idealism.
- (mathematics) Not actually present, but considered as present when limits at infinity are included.
Synonyms
- (optimal): best, ideal, optimal, optimum
- (flawless): see also Thesaurus:flawless
- (of ideas): conceptual, notional
- (existing only in mind): conceptual, imaginary
Derived terms
Related 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 ${displaystyle \mathbb }$ be the ring of integers and let ${displaystyle 2\mathbb }$ be its ideal of even integers. Then the quotient ring ${displaystyle \mathbb /2\mathbb }$ is a Boolean ring.
- The product of two ideals ${displaystyle {mathfrak }}$ and ${displaystyle {mathfrak }}$ is an ideal ${displaystyle {mathfrak }}$ which is a subset of the intersection of ${displaystyle {mathfrak }}$ and ${displaystyle {mathfrak }}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of ${displaystyle {mathfrak }}$ and ${displaystyle {mathfrak }}$ is a subset of ${displaystyle {mathfrak {a+b}}}$.
- (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.
- 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 ${displaystyle I}$ of a given set ${displaystyle X}$ is a nonempty subset of the powerset ${displaystyle {mathcal }(X)}$ such that: ${displaystyle (1)\ \emptyset \in I}$, ${displaystyle (2)\ A\in I\land B\subseteq A\implies B\in I}$ and ${displaystyle (3)\ A,B\in I\implies A\cup B\in I}$.
- (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.
- (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
- (order theory): order ideal
- (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 ideālis.
Pronunciation
- IPA^{(key)}: /ideˈal/, [i.ð̞eˈal]
Adjective
ideal (epicene, plural ideales)
- ideal
Noun
ideal m (plural ideales)
- ideal
Catalan
Etymology
Borrowed from Latin ideālis.
Pronunciation
- (Balearic, Central, Valencian) IPA^{(key)}: /i.deˈal/
Adjective
ideal m or f (masculine and feminine plural ideals)
- ideal
Derived terms
- idealment
Noun
ideal m (plural ideals)
- ideal
Galician
Etymology
From Latin ideālis.
Pronunciation
Adjective
ideal m or f (plural ideais)
- ideal
Derived terms
- idealmente
Noun
ideal m (plural ideais)
- ideal
German
Etymology
Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.
Pronunciation
- IPA^{(key)}: /ideˈaːl/
- Rhymes: -aːl
Adjective
ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) 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 ideālis.
Pronunciation
- IPA^{(key)}: [ˈɪdeal]
- 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, Jakarta: Language Development and Fostering Agency — Ministry of Education, Culture, Research, and Technology of the Republic Indonesia, 2016.
Luxembourgish
Pronunciation
- IPA^{(key)}: /ideˈaːl/, [idəˈaːl], /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
- “ideal” in The Bokmål Dictionary.
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
- “ideal” in The Nynorsk Dictionary.
Piedmontese
Pronunciation
- IPA^{(key)}: /ideˈal/
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
- idealmente
Noun
ideal m (plural ideais)
- ideal
- fantasy
Romanian
Etymology
Borrowed from French idéal, from Latin idealis.
Adjective
ideal m or n (feminine singular ideală, masculine plural ideali, feminine and neuter plural ideale)
- ideal
Declension
Related terms
- idealism
- idealist
- idealiza
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/ [i.ð̞eˈal]
- 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
- “ideal”, in Diccionario de la lengua española, Vigésima tercera edición, Real Academia Española, 2014
Swedish
Pronunciation
Noun
ideal n
- ideal; perfect standard
- (mathematics) ideal; special subsets of a ring
Declension
Anagrams
- ilade
Tagalog
Alternative forms
- ideyal
Etymology
Borrowed from Spanish ideal.
Pronunciation
- Hyphenation: i‧de‧al
- IPA^{(key)}: /ʔideˈal/, [ʔɪ.dɛˈal]
Noun
ideál
- ideal (model to be imitated)
- Synonyms: huwaran, uliran
- ambition
- Synonyms: mithi, mithiin, pangarap, ambisyon
Related terms
- idea
- idealismo
- idealista
Adjective
ideál
- 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
- ülkü