ideal
Part of speech: adj
noun
Morphological type: derivation
Topics: algebra
mathematics
order-theory
sciences
set-theory
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.
Inflected forms:
form: more ideal
pos: adj
tags: comparative
form: most ideal
pos: adj
tags: superlative
form: ideals
pos: noun
tags: plural
Senses:
gloss: Pertaining to ideas, or to a given idea.
pos: adj
gloss: Existing only in the mind; conceptual, imaginary.
pos: adj
examples: The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
At first, he began to doubt the reality of his adventures, but the acute pain in his shoulders when he attempted to rise, assured him that the kicking of the goblins was certainly not ideal.
gloss: Optimal; being the best possibility.
pos: adj
gloss: Perfect, flawless, having no defects.
pos: adj
examples: 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; […] .
gloss: Teaching or relating to the doctrine of idealism.
pos: adj
examples: the ideal theory or philosophy
gloss: Not actually present, but considered as present when limits at infinity are included.
pos: adj
examples: ideal point
An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.
gloss: A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
pos: noun
examples: 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.
With great humility, I call upon all Americans to help me keep our nation united in defense of those ideals which have been so eloquently proclaimed by Franklin Roosevelt. I want in turn to assure my fellow Americans and all of those who love peace and liberty throughout the world that I will support and defend those ideals with all my strength and all my heart.
gloss: A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
pos: noun
examples: 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;.
In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.
If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.
However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.
gloss: A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
pos: noun
examples: 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.
1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.
Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.
gloss: 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.
pos: noun
examples: 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.
gloss: 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.
pos: noun
examples: If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.
What really put primitive ideals in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such ideal is the annihilator of a very special kind of simple module, namely a highest weight module.
Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).
gloss: 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.
pos: noun
examples: 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.
Synonyms: order ideal
Antonyms: filter
Related words: idealism
idealistic
ideality
idealize
idealless
ideally
Derived words: 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
Translations:
lang: German
word: Idealzustand
lang: Polish
word: ideał
lang: Russian
word: идеа́л
roman: ideál