Definition of "ideal" in 英语

adjective

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

2. Existing only in the mind; conceptual, imaginary.

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. (mathematics) 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.

noun

1. 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

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

   • 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119, 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.

4. (set theory) 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.