Ideal

Adjective

  • Teaching or relating to the doctrine of idealism.

    "the ideal theory or philosophy"

  • Optimal; being the best possibility.
  • Existing only in the mind; conceptual, imaginary.
  • (mathematics) Not actually present, but considered as present when limits at infinity are included.

    "An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle."

  • Pertaining to ideas, or to a given idea.
  • (mathematics) Not actually present, but considered as present when limits at infinity are included.

    "ideal point"

  • Perfect, flawless, having no defects.

Noun

  • ‚ÄčA 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"

  • (for example, algebra) A subring closed under multiplication by its containing ring.

    "Let be the ring of integers and let be its ideal of even integers. Then the quotient ring is a Boolean ring.The product of two ideals and is an ideal which is a subset of the intersection of and . This should help to understand why maximal ideals are prime ideals. Likewise, the union of and is a subset of ."

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

    "If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an ideal."

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