Definition of IDEAL

ideal

Plural: ideals

Noun

  • A standard of perfection; a goal or principle to aspire to.
  • the idea of something that is perfect; something that one hopes to attain
  • model of excellence or perfection of a kind; one having no equal
  • A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
  • A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
  • A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
  • 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.
  • 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.
  • 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.

Adjective Satellite

  • conforming to an ultimate standard of perfection or excellence; embodying an ideal
  • constituting or existing only in the form of an idea or mental image or conception
    • "a poem or essay may be typical of its period in idea or ideal content"

Adjective

  • of or relating to the philosophical doctrine of the reality of ideas

Adj

  • Pertaining to ideas, or to a given idea.
  • Existing only in the mind; conceptual, imaginary.
  • Optimal; being the best possibility.
  • Perfect, flawless, having no defects.
  • Teaching or relating to the doctrine of idealism.
  • Not actually present, but considered as present when limits at infinity are included.

Examples

  • An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.
  • 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.
  • His gameplay was far from ideal, but he still managed to win by sheer luck.
  • ideal point
  • 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 ideal theory or philosophy
  • 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;.
  • 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.

Origin / 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.

Antonyms

filter

Scrabble Score: 6

ideal: valid Scrabble (US) TWL Word
ideal: valid Scrabble Word in Merriam-Webster MW Dictionary
ideal: valid Scrabble Word in International Collins CSW Dictionary

Words With Friends Score: 7

ideal: valid Words With Friends Word