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ideal

adj
/aɪˈdɪəl/UK/aɪˈdi(ə)l/US/ˈajɖɪjəl/

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.

  1. derived from idea — “idea”
  2. derived from ideālis — “existing in idea”
  3. derived from idéal

Definitions

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

  2. Existing only in the mind

    Existing only in the mind; conceptual, imaginary.

    • The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
    • Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
    • 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

    Optimal; being the best possibility.

  4. ›+ 12 more definitions

    1. 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; […] .

    2. Teaching or relating to the doctrine of idealism.

      • the ideal theory or philosophy

    3. 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.

    4. 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

    5. A two-sided ideal

      A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.

      • Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.
      • In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.

    6. A non-empty lower set (of a partially ordered set) which is closed under binary suprema…

      A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).

      • 1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
      • Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.

    7. A collection of sets, considered small or negligible, such that every subset of each…

      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.

    8. A Lie subalgebra (subspace that is closed under the Lie bracket) 𝖍 of a given Lie algebra…

      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.

      • If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
      • 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.
      • Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).

    9. A subsemigroup with the property that if any semigroup element outside of it is added to…

      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.

    10. A city in Georgia, United States.

    11. An unincorporated community in Illinois.

    12. An unincorporated community in South Dakota.

The neighborhood

Derived

coideal, counterideal, hyperideal, ideal gas, ideal gas law, ideal mixture, idealness, ideal number, ideal point, ideal polyhedron, ideal-seeking behavior, ideal solution, ideal triangle, ideal-typical, idele, nonideal, non-ideal gas, unideal, different ideal, 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

Vish — recursive loop

A definitional loop anchored at ideal. Each word in the ring is defined by the next; follow the chain far enough and it folds back on itself. Scroll to it and watch.

01ideal02mind03remember04image05external06company07professionally08professional09profession10faith

A definitional loop anchored at ideal. Each word in the ring appears in the definition of the next; follow the chain far enough and it folds back on itself.

10 hops · closes at ideal

curated · pre-corpus. live cycle detection across the full graph is the next major milestone.

sense glosses and etymology drawn from English Wiktionary · source · CC-BY-SA