ideal

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.

• Rhymes: -iːəl
• IPA^(key): /aɪˈdiːl/, /aɪˈdɪəl/, /aɪˈdiː.əl/

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ideal (comparative more ideal, superlative most ideal)

1. Pertaining to ideas, or to a given idea.
2. 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.

3. Optimal; being the best possibility.
4. 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; […] .

5. Teaching or relating to the doctrine of idealism.

   the ideal theory or philosophy

6. (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.

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

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

   • 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.‎^[3], 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.

3. (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 ${\displaystyle \mathbb {Z} }$ be the ring of integers and let ${\displaystyle 2\mathbb {Z} }$ be its ideal of even integers. Then the quotient ring ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ is a Boolean ring.

   The product of two ideals ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ is an ideal ${\displaystyle {\mathfrak {ab}}}$ which is a subset of the intersection of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ is a subset of ${\displaystyle {\mathfrak {a+b}}}$.

   • 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.

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).^[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^[4], 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.

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 ${\displaystyle {\mathcal {P}}(X)}$ 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.
   • 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 ${\displaystyle L}$ be an arbitrary semisimple Lie algebra. Then ${\displaystyle L}$ can be written uniquely as a direct sum ${\displaystyle L_{1}\oplus \dots \oplus L_{t}}$ of simple ideals (Theorem 5.2).

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.^[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.

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

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

• (mathematics): maximal ideal, principal ideal

• 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

• Delia, Elida, ailed, ladie

From Latin ideālis.

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

ideal (epicene, plural ideales)

1. ideal

ideal m (plural ideales)

1. ideal

• “ideal” in Diccionariu de la llingua asturiana (1ª edición). Academia de la Llingua Asturiana (2000). →ISBN.

Borrowed from Latin ideālis.

• IPA^(key): (Central, Balearic, Valencia) [i.ðeˈal]

ideal m or f (masculine and feminine plural ideals)

1. ideal

• idealisme
• idealista
• idealitzar
• idealment

ideal m (plural ideals)

1. ideal

From Latin ideālis.

• IPA^(key): /idɛˈal/ [i.ð̞ɛˈɑɫ]
• Rhymes: -al
• Hyphenation: i‧de‧al

ideal m or f (plural ideais)

1. ideal

• idealmente

ideal m (plural ideais)

1. ideal

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

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

• IPA^(key): /ideˈaːl/
• Rhymes: -aːl

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

1. ideal (optimal, perfect)

• idealerweise

• “ideal” in Duden online
• “ideal” in Digitales Wörterbuch der deutschen Sprache

From Dutch ideaal, from French idéal, from Medieval Latin ideālis.

• (Standard Indonesian) IPA^(key): /iˈdeal/ [iˈde.al]
• Rhymes: -eal
• Syllabification: i‧de‧al

idéal (comparative lebih ideal, superlative paling ideal)

1. ideal; optimal, being the best possibility
2. ideal; pertaining to ideas, or to a given idea

idéal (plural ideal-ideal)

1. (mathematics) ideal; a subring closed under multiplication by its containing ring

• idiil (noun)

• “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.

• IPA^(key): /ideˈaːl/, [idəˈaːl], /idiˈaːl/

ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)

1. ideal

From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).

ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene)

1. ideal

• “ideal” in The Bokmål Dictionary.

From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”).

ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)

1. ideal

• “ideal” in The Nynorsk Dictionary.

• IPA^(key): /ideˈal/

ideal

1. ideal

ideal m

1. ideal

Borrowed from Latin ideālis.

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

ideal m or f (plural ideais)

1. ideal
2. notional

• idealmente

ideal m (plural ideais)

1. ideal
2. fantasy

Borrowed from French idéal, from Latin idealis.

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

1. ideal

• idealism
• idealist
• idealiza

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

idèāl m inan (Cyrillic spelling идѐа̄л)

1. ideal

From Latin ideālis.

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

ideal m or f (masculine and feminine plural ideales)

1. ideal

ideal m (plural ideales)

1. ideal

• “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], 10 December 2024

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ideal (not comparable)

1. ideal (perfect)

   Synonym: idealisk

   under ideala förhållanden

   under ideal conditions

ideal n

1. ideal; perfect standard
2. (mathematics) ideal; special subsets of a ring

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

• ilade

• ideyal

Borrowed from Spanish ideal.

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

ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

1. ideal (model to be imitated)

   Synonyms: huwaran, uliran

2. ambition

   Synonyms: mithi, mithiin, pangarap, ambisyon

• idea
• idealismo
• idealista

ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔)

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

   Synonym: uliran

Borrowed from French idéal.

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

ideal

1. ideal

ideal (definite accusative ideali, plural idealler)

1. ideal

• ülkü

• ideol

Borrowed from English ideal.^[1]

• IPA^(key): /ɪˈdɛal/

ideal (feminine singular ideal, plural ideal, not comparable)

1. ideal^[1][2]

   Synonym: delfrydol

ideal m (plural idealau)

1. ideal^[1][2]

   Synonym: delfryd