ideal

Etymology
From, from , , from ; see. In mathematics, the noun ring theory sense was first introduced by German mathematician 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.

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.
 * 5) * 1751 April 13,, , 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;.
 * 1) Teaching or relating to the doctrine of idealism.
 * 2)  Not actually present, but considered as present when limits at infinity are included.
 * There will always be a wide interval between practical and ideal excellence;.
 * 1) Teaching or relating to the doctrine of idealism.
 * 2)  Not actually present, but considered as present when limits at infinity are included.
 * 1)  Not actually present, but considered as present when limits at infinity are included.

Synonyms

 * see also Thesaurus:flawless
 * see also Thesaurus:flawless

Translations

 * Afrikaans: voorbeeldig, ideaal
 * Arabic: مِثَالِيّ
 * Asturian: ideal
 * Belarusian: ідэа́льны
 * Bulgarian:
 * Catalan:
 * Chinese:
 * Mandarin:
 * Czech:
 * Dutch: ,
 * Esperanto: ideala
 * Finnish:, , ,
 * French:
 * Galician: ideal
 * German:, bestmöglich,
 * Hindi:, ,
 * Hungarian:
 * Ido:
 * Irish: idéalach
 * Italian:
 * Japanese: ,
 * Korean:, 리상적(理想的)
 * Latvian: ideāls
 * Luxembourgish: ideal
 * Macedonian: идеа́лен
 * Malayalam:
 * Norman: idéal
 * Persian:, ایده‌آل
 * Polish:
 * Portuguese:
 * Russian:
 * Slovak: ideálny
 * Slovene: idealen
 * Spanish:
 * Telugu:
 * Ukrainian: ідеа́льний


 * Afrikaans: ideaal
 * Asturian: ideal
 * Belarusian: ідэа́льны, даскана́лы
 * Bulgarian: ,
 * Catalan:
 * Czech:
 * Dutch:, ,
 * Esperanto: ideala
 * Finnish: ,
 * French: ,
 * Galician: ideal
 * German:, , , ,
 * Greek:
 * Hindi:, ,
 * Hungarian: ,
 * Ido:
 * Irish: idéalach
 * Manx: slanjeant
 * Maori: takarepakore, paruhi
 * Norman: idéal
 * Persian:, فربود,
 * Polish:
 * Portuguese:
 * Russian: ,
 * Slovene: idealen
 * Spanish:
 * Telugu:
 * Ukrainian: ідеа́льний, доскона́лий


 * Afrikaans: ideëel, denkbeeldig
 * Asturian: ideal
 * Bulgarian: ,
 * Catalan:
 * Dutch: ,
 * Esperanto: ideala
 * Finnish:
 * French:
 * Galician: ideal
 * German:
 * Hebrew: אידיאלי
 * Hungarian:
 * Ido:
 * Malayalam:
 * Manx: ard-smooinagh, ard-smooinaghtagh
 * Norman: idéal
 * Polish: ,
 * Portuguese:
 * Russian:, ,
 * Slovene: idejen
 * Spanish:


 * Persian:


 * Esperanto:
 * Indonesian:
 * Interlingua:
 * Italian:
 * Spanish:

Noun

 * 1) A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.
 * 2) 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
 * 1)  A subring closed under multiplication by its containing ring.
 * Let $$\mathbb{Z}$$ be the ring of integers and let $$2\mathbb{Z}$$ be its ideal of even integers. Then the quotient ring $$\mathbb{Z} / 2\mathbb{Z}$$ is a Boolean ring.
 * The product of two ideals $$\mathfrak{a}$$ and $$\mathfrak{b}$$ is an ideal $$\mathfrak{a b}$$ which is a subset of the intersection of $$\mathfrak{a}$$ and $$\mathfrak{b}$$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of $$\mathfrak{a}$$ and $$\mathfrak{b}$$ is a subset of $$\mathfrak{a + b}$$.
 * 1)  A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
 * 2) * 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II,, 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.
 * 1)  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 $$\mathcal{P}(X)$$ such that: $$(1)\ \emptyset \in I$$, $$(2)\ A \in I \and B \subseteq A\implies B\in I$$ and $$(3)\ A,B \in I\implies A\cup B \in I$$.
 * 1)  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.
 * 2)  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.
 * 1)  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 $$\mathcal{P}(X)$$ such that: $$(1)\ \emptyset \in I$$, $$(2)\ A \in I \and B \subseteq A\implies B\in I$$ and $$(3)\ A,B \in I\implies A\cup B \in I$$.
 * 1)  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.
 * 2)  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.
 * 1)  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.
 * 1)  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.

Hyponyms

 * maximal ideal, principal ideal

Translations

 * Afrikaans: ideaal
 * Asturian: ideal
 * Belarusian: ідэа́л
 * Bulgarian:
 * Catalan:
 * Chinese:
 * Mandarin: ,
 * Czech: ideál
 * Dutch:, ,
 * Esperanto: ideala, perfekta
 * Finnish: ,
 * French:
 * Galician: ideal
 * German:
 * Greek:
 * Hebrew:
 * Hindi:, ,
 * Hungarian:, ,
 * Ido:
 * Indonesian:
 * Interlingua: ideal
 * Irish: idéal
 * Italian:
 * Japanese:
 * Kazakh: мұрат
 * Korean: ,
 * Persian:, ایدئال
 * Polish:
 * Portuguese:
 * Punjabi:
 * Russian:
 * Slovak: ideál
 * Slovene: ideal
 * Spanish:
 * Swedish:
 * Telugu:
 * Ukrainian: ідеа́л
 * Vietnamese:
 * Welsh:


 * Chinese:
 * Mandarin:
 * Dutch:
 * Finnish:
 * French:
 * German:
 * Kazakh: идеал
 * Polish:
 * Romanian:
 * Russian:
 * Swedish:


 * Chinese:
 * Mandarin:
 * Irish: idéal
 * Russian:


 * German:
 * Polish:
 * Russian:

Etymology
From.

Etymology
.

Etymology
From.

Etymology
, from. .

Adjective

 * 1) ideal optimal, perfect

Etymology
From, from , from.

Adjective

 * 1) optimal; being the best possibility.
 * 2) pertaining to ideas, or to a given idea.
 * 1) pertaining to ideas, or to a given idea.

Noun

 * : a subring closed under multiplication by its containing ring.

Adjective

 * 1) ideal

Etymology
From, from , from.

Etymology
From, from , from.

Etymology
.

Adjective

 * 1) notional
 * 1) notional

Noun

 * 1) fantasy
 * 1) fantasy

Etymology
, from.

Etymology
From.

Adjective

 * 1)  (perfect)

Noun

 * 1) ; perfect standard
 * 2)  ; special subsets of a ring

Etymology
.

Noun

 * 1) ambition
 * 1) ambition
 * 1) ambition

Adjective

 * 1) worthy of imitation
 * 1) worthy of imitation

Etymology
Borrowed from.

Adjective

 * 1) ideal

Noun

 * 1) ideal