ideal

Meanings

1. 1
   adj

   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

2. 2
   adj

   conforming to an ultimate standard of perfection or excellence

   embodying an ideal

3. 3
   adj

   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 —

4. 4
   adj

   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. 5
   adj

   Teaching or relating to the doctrine of idealism.

   the ideal theory or philosophy

6. 6
   adj

   Not actually present, but considered as present when limits at infinity are included.

   ideal point

7. 7
   noun

   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

8. 8
   noun

   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.

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.