ideal

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)

1. Optimal; being the best possibility.
2. 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; […] .

3. Pertaining to ideas, or to a given idea.
4. Existing only in the mind; conceptual, imaginary.
5. Teaching or relating to the doctrine of idealism.

6. (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)

2. 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

3. (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 ${displaystyle {mathfrak {a+b}}}$.
4. (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.
5. (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 ${displaystyle (3)\ A,B\in I\implies A\cup B\in I}$.

6. (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.
7. (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)

1. ideal

Noun

ideal m (plural ideales)

1. 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)

1. ideal

Derived terms

• idealment

Noun

ideal m (plural ideals)

1. ideal

Galician

Etymology

From Latin ideālis.

Pronunciation

Adjective

ideal m or f (plural ideais)

1. ideal

Derived terms

• idealmente

Noun

ideal m (plural ideais)

1. 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)

1. 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

1. ideal:
   1. optimal; being the best possibility.
   2. 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)

1. (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)

1. 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)

1. 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)

1. ideal

References

• “ideal” in The Nynorsk Dictionary.

Piedmontese

Pronunciation

• IPA^(key): /ideˈal/

Adjective

ideal

1. ideal

Noun

ideal m

1. ideal

Portuguese

Etymology

Borrowed from Latin ideālis.

Pronunciation

• Rhymes: -al, -aw
• Hyphenation: i‧de‧al

Adjective

ideal m or f (plural ideais)

1. ideal
2. notional

Derived terms

• idealmente

Noun

ideal m (plural ideais)

1. ideal
2. 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)

1. ideal

Declension

Related terms

• idealism
• idealist
• idealiza

Serbo-Croatian

Pronunciation

• IPA^(key): /iděaːl/
• Hyphenation: i‧de‧al

Noun

idèāl m (Cyrillic spelling идѐа̄л)

1. ideal

Declension

Spanish

Etymology

From Latin ideālis.

Pronunciation

• IPA^(key): /ideˈal/ [i.ð̞eˈal]
• Rhymes: -al
• Syllabification: i‧de‧al

Adjective

ideal (plural ideales)

1. ideal

Derived terms

Noun

ideal m (plural ideales)

1. ideal

Further reading

• “ideal”, in Diccionario de la lengua española, Vigésima tercera edición, Real Academia Española, 2014

Swedish

Pronunciation

Noun

ideal n

1. ideal; perfect standard
2. (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): /ʔideˈjal/, [ʔɪ.dɛˈjal]

Noun

ideál

1. ideal (model to be imitated)

   Synonyms: huwaran, uliran

2. ambition

   Synonyms: mithi, mithiin, pangarap, ambisyon

Related terms

• idea
• idealismo
• idealista

Adjective

ideál

1. ideal (just as what one wants)
2. worthy of imitation

   Synonym: uliran

Turkish

Etymology

Borrowed from French idéal.

Pronunciation

• IPA^(key): /ideˈɑl/
• Hyphenation: i‧de‧al

Adjective

ideal

1. ideal

Noun

ideal (definite accusative ideali, plural idealler)

1. ideal

Declension

Synonyms

• ülkü