IDEAL

ideal

(adjective) 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”

ideal

(adjective) conforming to an ultimate standard of perfection or excellence; embodying an ideal

ideal, idealistic

(adjective) of or relating to the philosophical doctrine of the reality of ideas

ideal

(noun) the idea of something that is perfect; something that one hopes to attain

ideal, paragon, nonpareil, saint, apotheosis, nonesuch, nonsuch

(noun) model of excellence or perfection of a kind; one having no equal

Source: WordNet® 3.1

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Proper noun

Ideal

A city in Georgia, United States.

An unincorporated community in Illinois.

An unincorporated community in South Dakota.

Anagrams

• Delia, Elida, ailed, ladie

Etymology

Adjective

ideal (comparative more ideal, superlative most ideal)

Optimal; being the best possibility.

Perfect, flawless, having no defects.

Pertaining to ideas, or to a given idea.

Existing only in the mind; conceptual, imaginary.

Teaching or relating to the doctrine of idealism.

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

Synonyms

• See also flawless

Noun

ideal (plural ideals)

A perfect standard of beauty, intellect etc, or a standard of excellence to aim at.

(algebra, ring theory) A subring closed under multiplication by its containing ring.

Let $\backslash (\backslash mathbb\{Z\}\backslash )$ be the ring of integers and let $\backslash (2\backslash mathbb\{Z\}\backslash )$ be its ideal of even integers. Then the quotient ring $\backslash (\backslash mathbb\{Z\}\; /\; 2\backslash mathbb\{Z\}\backslash )$ is a Boolean ring.

The product of two ideals $\backslash (\backslash mathfrak\{a\}\backslash )$ and $\backslash (\backslash mathfrak\{b\}\backslash )$ is an ideal $\backslash (\backslash mathfrak\{a\; b\}\backslash )$ which is a subset of the intersection of $\backslash (\backslash mathfrak\{a\}\backslash )$ and $\backslash (\backslash mathfrak\{b\}\backslash )$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of $\backslash (\backslash mathfrak\{a\}\backslash )$ and $\backslash (\backslash mathfrak\{b\}\backslash )$ is a subset of $\backslash (\backslash mathfrak\{a\; +\; b\}\backslash )$.

(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).

(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.

Formally, an ideal $\backslash (I\backslash )$ of a given set $\backslash (X\backslash )$ is a nonempty subset of the powerset $\backslash (\backslash mathcal\{P\}(X)\backslash )$ such that: $\backslash ((1)\backslash \; \backslash emptyset\; \backslash in\; I\backslash )$, $\backslash ((2)\backslash \; A\; \backslash in\; I\; \backslash and\; B\; \backslash subseteq\; A\backslash implies\; B\backslash in\; I\backslash )$ and $\backslash ((3)\backslash \; A,B\; \backslash in\; I\backslash implies\; A\backslash cup\; B\; \backslash in\; I\backslash )$.

(algebra, Lie theory) A Lie subalgebra (subspace that is closed under the Lie bracket) ? of a given Lie algebra ? such that the Lie bracket [?,?] is a subset of ?.

Synonyms

• (type of Lie subalgebra): Lie ideal

Antonyms

• (order theory): filter

Hyponyms

• (mathematics): maximal ideal, principal ideal

Anagrams

• Delia, Elida, ailed, ladie

Source: Wiktionary

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I*de"al, a. Etym: [L. idealis: cf. F. idéal.]

1. Existing in idea or thought; conceptional; intellectual; mental; as, ideal knowledge.

2. Reaching an imaginary standard of excellence; fit for a model; faultless; as, ideal beauty. Byron. There will always be a wide interval between practical and ideal excellence. Rambler.

3. Existing in fancy or imagination only; visionary; unreal. "Planning ideal common wealth." Southey.

4. Teaching the doctrine of idealism; as, the ideal theory or philosophy.

5. (Math.)

Definition: Imaginary.

Syn.

– Intellectual; mental; visionary; fanciful; imaginary; unreal; impracticable; utopian.

I*de"al, n.

Definition: A mental conception regarded as a standard of perfection; a model of excellence, beauty, etc. The ideal is to be attained by selecting and assembling in one whole the beauties and perfections which are usually seen in different individuals, excluding everything defective or unseemly, so as to form a type or model of the species. Thus, the Apollo Belvedere is the ideal of the beauty and proportion of the human frame. Fleming. Beau ideal. See Beau ideal.

Source: Webster’s Unabridged Dictionary 1913 Edition

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