# ideal
## English
### Etymology

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea +‎ -al, 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: /aɪˈdiːl/, /aɪˈdɪəl/, /aɪˈdiː.əl/
* Audio (US):(file)

### 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 —
** 1816 June – 1817 April/May (date written), [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, published 1 January 1818, →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

* (optimal): best, ideal, optimal, optimum
* (flawless): see also Thesaurus:flawless
* (of ideas): conceptual, notional
* (existing only in mind): conceptual, imaginary

#### Derived terms

* ideal gas
* ideal gas law
* ideal mixture
* ideal number
* ideal point
* ideal polyhedron
* ideal-seeking behavior
* ideal solution
* ideal triangle
* ideal-typical
* non-ideal gas

#### Related terms

* idealism
* idealistic
* ideality
* idealize
* idealless
* ideally

#### Translations

    The translations below need to be checked and inserted above into the appropriate translation tables. See instructions at Wiktionary:Entry layout § Translations.

### 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 two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
    Let $\mathbb {Z}$ be the ring of integers and let $2\mathbb {Z}$ be its ideal of even integers. Then the quotient ring $\mathbb {Z} /2\mathbb {Z}$ is a Boolean ring.
    The product of two ideals ${\mathfrak {a}}$ and ${\mathfrak {b}}$ is an ideal ${\mathfrak {ab}}$ which is a subset of the intersection of ${\mathfrak {a}}$ and ${\mathfrak {b}}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of ${\mathfrak {a}}$ and ${\mathfrak {b}}$ is a subset of ${\mathfrak {a+b}}$.
** 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 [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.
* (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 $I$ of a given set $X$ is a nonempty subset of the powerset ${\mathcal {P}}(X)$ such that: $(1)\ \emptyset \in I$, $(2)\ A\in I\land B\subseteq A\implies B\in I$ and $(3)\ A,B\in I\implies A\cup B\in I$.
* (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.
** 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 $L$ be an arbitrary semisimple Lie algebra. Then $L$ can be written uniquely as a direct sum $L_{1}\oplus \dots \oplus L_{t}$ 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

* (order theory): order ideal
* (type of Lie subalgebra): Lie ideal

#### Antonyms

* (antonym(s) of “order theory”): filter

#### Hyponyms

* (mathematics): maximal ideal, principal ideal

#### Derived terms

* factor ideal
* fractional ideal
* Frink ideal
* ideal lattice
* lattice ideal
* left ideal
* minimal ideal
* Platonic ideal
* primary ideal
* prime ideal
* principal ideal
* principal ideal domain
* principal ideal ring
* pseudoideal
* radical ideal
* right ideal
* subideal
* two-sided ideal
* vanishing ideal
* zero ideal

#### Translations
### References

* ↑ Boolean prime ideal theorem on Wikipedia.Wikipedia
* ↑ Vaughan Pratt (2004) Chapter 1 : Lattice Theory‎, boole.stanford.edu, §1.3.5

### 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 on Wikipedia.Wikipedia

### Anagrams

* Delia, Elida, ailed, ladie

## Asturian
### Etymology

From Latin ideālis.

### Pronunciation

* IPA: /ideˈal/ [i.ð̞eˈal]

### Adjective

ideal (epicene, plural ideales)
* ideal

### Noun

ideal m (plural ideales)
* ideal

## Catalan
### Etymology

Borrowed from Latin ideālis.

### Pronunciation

* IPA: (Central, Balearic, Valencia) [i.ðeˈal]

### Adjective

ideal m or f (masculine and feminine plural ideals)
* ideal

#### Derived terms

* idealisme
* idealista
* idealitzar
* idealment

### Noun

ideal m (plural ideals)
* ideal

## Galician
### Etymology

From Latin ideālis.

### Pronunciation

* IPA: /idɛˈal/ [i.ð̞ɛˈɑɫ]
* Rhymes: -al
* Hyphenation: i‧de‧al

### Adjective

ideal m or f (plural ideais)
* ideal

#### Derived terms

* idealmente

### Noun

ideal m (plural ideais)
* ideal

### Further reading

* “ideal”, in Dicionario da Real Academia Galega (in Galician), A Coruña: Royal Galician Academy, 2012–2025
* “ideal” in Dicionário Estraviz de galego (2014).

## German
### Etymology

Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.

### Pronunciation

* IPA: /ideˈaːl/
* Rhymes: -aːl
* Audio:(file)

### Adjective

ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten)
* ideal (optimal, perfect)

#### 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: [ˈɪdeal]
* Hyphenation: idé‧al

### Adjective

ideal
* ideal:
** optimal; being the best possibility
** pertaining to ideas, or to a given idea

### Noun

ideal (plural)
* (mathematics) ideal: a subring closed under multiplication by its containing ring

#### Alternative forms

* idiil

### Affixed terms

* mengidealkan

### Related terms

* ide

### Further reading

* “ideal” in Kamus Besar Bahasa Indonesia, Jakarta: Agency for Language Development and Cultivation – Ministry of Education, Culture, Research, and Technology of the Republic of Indonesia, 2016.

## Luxembourgish
### Pronunciation

* IPA: /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: /ideˈal/

### Adjective

ideal
* ideal

### Noun

ideal m
* ideal

## Portuguese
### Etymology

Borrowed from Latin ideālis.

### Pronunciation

* (Brazil) IPA: /i.deˈaw/ [i.deˈaʊ̯]
* (Portugal) IPA: /iˈdjal/ [iˈðjaɫ]
* (Portugal) IPA: /iˈdjal/ [iˈðjaɫ]
** (Southern Portugal) IPA: /iˈdja.li/ [iˈðja.li]
* 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.

### Pronunciation

* Audio:(file)

### Adjective

ideal m or n (feminine singular ideală, masculine plural ideali, feminine and neuter plural ideale)
* ideal

#### Related terms

* idealism
* idealist
* idealiza

## Serbo-Croatian
### Pronunciation

* IPA: /iděaːl/
* Hyphenation: i‧de‧al

### Noun

idèāl m (Cyrillic spelling идѐа̄л)
* ideal

#### Declension
## Spanish
### Etymology

From Latin ideālis.

### Pronunciation

* IPA: /ideˈal/ [i.ð̞eˈal]
* Rhymes: -al
* Syllabification: i‧de‧al

### Adjective

ideal m or f (masculine and feminine plural ideales)
* ideal

#### Derived terms

* idealidad
* idealizar
* idealmente

### Noun

ideal m (plural ideales)
* ideal

### Further reading

* “ideal”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), online version 23.8, Royal Spanish Academy [Spanish: Real Academia Española], 2024 December 10

## Swedish
### Pronunciation

* Audio:(file)

### Adjective

ideal (not comparable)
* ideal (perfect)
    Synonym: idealisk
    under ideala förhållandenunder ideal conditions

#### Declension

$^1$ The indefinite superlative forms are only used in the predicative. $^2$ Dated or archaic. $^3$ Only used, optionally, to refer to things whose natural gender is masculine.

### Noun

ideal n
* ideal; perfect standard
* (mathematics) ideal; special subsets of a ring

#### Related terms

* idé
* idealisera
* idealisk
* idealism
* idealist
* idealistisk

### References

* ideal in Svensk ordbok (SO)
* ideal in Svenska Akademiens ordlista (SAOL)
* ideal in Svenska Akademiens ordbok (SAOB)

### Anagrams

* ilade

## Tagalog
### Alternative forms

* ideyal

### Etymology

Borrowed from Spanish ideal.

### Pronunciation

* (Standard Tagalog) IPA: /ʔideˈal/ [ʔɪ.d̪ɛˈal]
* Rhymes: -al
* Syllabification: i‧de‧al

### Noun

ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
* ideal (model to be imitated)
    Synonyms: huwaran, uliran
* ambition
    Synonyms: mithi, mithiin, pangarap, ambisyon

#### Related terms

* idea
* idealismo
* idealista

### Adjective

ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)
* ideal (just as what one wants)
* worthy of imitation
    Synonym: uliran

## Turkish
### Etymology

Borrowed from French idéal.

### Pronunciation

* IPA: /ideˈɑl/
* Hyphenation: i‧de‧al

### Adjective

ideal
* ideal

### Noun

ideal (definite accusative ideali, plural idealler)
* ideal

#### Declension

singular
plural
nominative
ideal
idealler
definite accusative
ideali
idealleri
dative
ideale
ideallere
locative
idealde
ideallerde
ablative
idealden
ideallerden
genitive
idealin
ideallerin
Possessive forms
nominative
singular
plural
1st singular
idealim
ideallerim
2nd singular
idealin
ideallerin
3rd singular
ideali
idealleri
1st plural
idealimiz
ideallerimiz
2nd plural
idealiniz
idealleriniz
3rd plural
idealleri
idealleri
definite accusative
singular
plural
1st singular
idealimi
ideallerimi
2nd singular
idealini
ideallerini
3rd singular
idealini
ideallerini
1st plural
idealimizi
ideallerimizi
2nd plural
idealinizi
ideallerinizi
3rd plural
ideallerini
ideallerini
dative
singular
plural
1st singular
idealime
ideallerime
2nd singular
idealine
ideallerine
3rd singular
idealine
ideallerine
1st plural
idealimize
ideallerimize
2nd plural
idealinize
ideallerinize
3rd plural
ideallerine
ideallerine
locative
singular
plural
1st singular
idealimde
ideallerimde
2nd singular
idealinde
ideallerinde
3rd singular
idealinde
ideallerinde
1st plural
idealimizde
ideallerimizde
2nd plural
idealinizde
ideallerinizde
3rd plural
ideallerinde
ideallerinde
ablative
singular
plural
1st singular
idealimden
ideallerimden
2nd singular
idealinden
ideallerinden
3rd singular
idealinden
ideallerinden
1st plural
idealimizden
ideallerimizden
2nd plural
idealinizden
ideallerinizden
3rd plural
ideallerinden
ideallerinden
genitive
singular
plural
1st singular
idealimin
ideallerimin
2nd singular
idealinin
ideallerinin
3rd singular
idealinin
ideallerinin
1st plural
idealimizin
ideallerimizin
2nd plural
idealinizin
ideallerinizin
3rd plural
ideallerinin
ideallerinin
Predicative forms
singular
plural
1st singular
idealim
ideallerim
2nd singular
idealsin
ideallersin
3rd singular
ideal idealdir
idealler ideallerdir
1st plural
idealiz
idealleriz
2nd plural
idealsiniz
ideallersiniz
3rd plural
idealler
ideallerdir :

#### Synonyms

* ülkü

## Welsh
### Alternative forms

* ideol

### Etymology

Borrowed from English ideal.

### Pronunciation

* IPA: /ɪˈdɛal/

### Adjective

ideal (feminine singular ideal, plural ideal, not comparable)
* ideal
    Synonym: delfrydol

### Noun

ideal m (plural idealau)
* ideal
    Synonym: delfryd

### Mutation

Note: Certain mutated forms of some words can never occur in standard Welsh. All possible mutated forms are displayed for convenience.

### References

* 1 2 3 R. J. Thomas, G. A. Bevan, P. J. Donovan, A. Hawke et al., editors (1950–present), “ideal”, in Geiriadur Prifysgol Cymru Online (in Welsh), University of Wales Centre for Advanced Welsh & Celtic Studies
* 1 2 Delyth Prys, J.P.M. Jones, Owain Davies, Gruffudd Prys (2006) Y Termiadur: termau wedi'u safoni; standardised terminology‎ (in Welsh), Cardiff: Awdurdod cymwysterau, cwricwlwm ac asesu Cymru (Qualifications curriculum & assessment authority for Wales), →ISBN