ideal

名词 n. 形容词 adj.
中文释义
英 /aɪˈdɪəl/    美 /aɪˈdi(ə)l/

英文释义

名词 n.
  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.
  2. 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.
  3. A 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.
  4. 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.
    — 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. 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.
    — If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
  6. 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.
    — 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.
形容词 adj.
  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 —
  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. Not actually present, but considered as present when limits at infinity are included.
    — ideal point

词形变化

more ideal comparative most ideal superlative ideals plural

词汇关系

近义词
order ideal
反义词
filter
下位词
maximal ideal principal ideal
衍生词
coideal counterideal hyperideal ideal gas ideal gas law ideal mixture idealness ideal number ideal point ideal polyhedron ideal-seeking behavior ideal solution ideal triangle ideal-typical idele nonideal non-ideal gas unideal different ideal factor ideal fractional ideal Frink ideal ideal lattice lattice ideal left ideal minimal ideal Platonic ideal primary ideal prime ideal principal ideal principal ideal domain principal ideal ring pseudoideal radical ideal right ideal subideal two-sided ideal vanishing ideal zero ideal
相关词
idealism idealistic ideality idealize idealless ideally

词源

词源 1
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.
词源 2
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.
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