ideal

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.

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

• Audio (US)

  (file)

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.
   • 1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:

     The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —

   • 1818, Mary Shelley, Frankenstein, or the Modern Prometheus, Chapter 4,

     Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.

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.

• See also Thesaurus:flawless

ideal (plural ideals)

1. 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

2. (algebra, ring theory) A subring closed under multiplication by its containing 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, 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.

3. (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, 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.

4. (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 ${\displaystyle (3)\ A,B\in I\implies A\cup B\in I}$ .

5. (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, Che-Young Lee (translator), Zhe-Xian Wan, 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, 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).

• (type of Lie subalgebra): Lie ideal

• (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#Subalgebras, ideals and homomorphisms on Wikipedia.Wikipedia

• Delia, Elida, ailed, ladie

From Latin ideālis.

ideal (epicene, plural ideales)

1. ideal

ideal m (plural ideales)

1. ideal

From Latin ideālis.

ideal (masculine and feminine plural ideals)

1. ideal

• idealment

ideal m (plural ideals)

1. ideal

From Latin ideālis.

ideal m or f (plural ideais)

1. ideal

• idealmente

ideal m (plural ideais)

1. ideal

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

• IPA^(key): /ideˈaːl/

• Audio

  (file)

• Rhymes: -aːl

ideal (comparative idealer, superlative am idealsten)

1. ideal (optimal, perfect)

• idealerweise

• ideal in Duden online

• 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. an 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. an ideal

• “ideal” in The Nynorsk Dictionary.

From Latin ideālis.

• (Portugal) IPA^(key): /iˈðjaɫ/
• (Brazil) IPA^(key): /i.deˈaw/
• Hyphenation: i‧de‧al

ideal m or f (plural ideais, comparable)

1. ideal
2. notional

• idealmente

ideal m (plural ideais)

1. ideal
2. fantasy

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

idèāl m (Cyrillic spelling идѐа̄л)

1. ideal

From Latin ideālis.

• IPA^(key): /ideˈal/, [iðeˈal]

ideal (plural ideales)

1. ideal

ideal m (plural ideales)

1. ideal

• audio

  (file)

ideal n

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

Borrowed from French idéal.

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

ideal (comparative daha ideal, superlative en ideal)

1. ideal

ideal (definite accusative ideali, plural idealler)

1. ideal

• ülkü