Idealname

• 1

  A city in Georgia, United States.

• 2

  An unincorporated community in Illinois.

• 3

  An unincorporated community in South Dakota.

idealadj/aɪˈdɪəl/

• 1

  Pertaining to ideas, or to a given idea.

• 2

  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.
• 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

  mathematicssciencesNot 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.

idealnoun/aɪˈdɪəl/

• 1

  A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.

  • Varvara Andreevna, when I was very young, I set before myself the ideal of the woman I loved and should be happy to call my wife.
  • 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.
• 2

  algebramathematicssciencesA 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.
  • The product of two ideals #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is an ideal #92;mathfrak#123;ab#125; which is a subset of the intersection of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125;. This should help to understand why maximal ideals are prime ideals. Likewise, the union of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is a subset of #92;mathfrak#123;a#43;b#125;.
• 3

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

  • 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.
  • 1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
• 4

  mathematicssciencesset-theoryA 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 #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.
• 5

  algebramathematicssciencesA 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.
• 6

  algebramathematicssciencesA 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.