Ideal

English

Etymology

From Template:Der, from Template:Der, Template:Surf, from Template:Der; see Template:M.
In mathematics, the noun ring theory sense was first introduced by German mathematician Template:W 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.

Pronunciation

• Template:Rhymes
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Adjective

1. Pertaining to ideas, or to a given idea.
2. Existing only in the mind; conceptual, imaginary.
   • Template:Quote-text
   • Template:RQ:Mary Shelley Frankenstein
   • Template:RQ:Dickens Pickwick Papers
3. Optimal; being the best possibility.
4. Perfect, flawless, having no defects.
   • 1751 April 13, Template:W, Template:W, 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;Template:....

5. Teaching or relating to the doctrine of idealism.

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6. Template:Lb Not actually present, but considered as present when limits at infinity are included.

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Synonyms

• Template:Sense Template:L, Template:L, Template:L, Template:L
• Template:Sense see also Thesaurus:flawless
• Template:Sense Template:L, Template:L
• Template:Sense Template:L, Template:L

Translations

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• Esperanto: Template:T-check
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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. Template:SenseidA 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

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3. Template:Lb A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.

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4. Template:Lb 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, Template:W, 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.

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5. Template:Lb 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.

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6. Template:Lb 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.
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7. Template:Lb 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.

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Synonyms

• Template:Sense Template:L
• Template:Sense Template:L

Antonyms

• Template:Antsense Template:L

Hyponyms

• Template:Sense maximal ideal, principal ideal

Translations

• Afrikaans: Template:T
• Asturian: Template:T
• Belarusian: Template:T
• Bulgarian: Template:T+
• Catalan: Template:T+
• Chinese:

  Mandarin: Template:T+, Template:T+

• Czech: Template:T
• Dutch: Template:T+, Template:T+, Template:T+
• Esperanto: Template:T, Template:T
• Finnish: Template:T+, Template:T+
• French: Template:T+
• Galician: Template:T
• German: Template:T+
• Greek: Template:T+
• Hebrew: Template:T+
• Hindi: Template:T+, Template:T+, Template:T+
• Hungarian: Template:T+, Template:T+, Template:T+
• Ido: Template:T+
• Indonesian: Template:T+
• Interlingua: Template:T
• Irish: Template:T
• Italian: Template:T+
• Japanese: Template:T+
• Kazakh: Template:T
• Korean: Template:T+, Template:T+ Template:Qualifier
• Persian: Template:T+, Template:T
• Polish: Template:T+
• Portuguese: Template:T+
• Punjabi: Template:T+
• Russian: Template:T+
• Slovak: Template:T
• Slovene: Template:T
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• Dutch: Template:T+
• Finnish: Template:T+
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• Japanese: Template:T
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Further reading

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Anagrams

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Asturian

Etymology

From Template:Der.

Adjective

1. Template:L

Noun

1. Template:L

Catalan

Etymology

Template:Bor+.

Pronunciation

• Template:Ca-IPA

Adjective

1. Template:L

Derived terms

• Template:L
• Template:L
• Template:L
• Template:L

Noun

1. Template:L

Galician

Etymology

From Template:Der.

Pronunciation

• Template:Hyph

Adjective

1. Template:L

Derived terms

• Template:L

Noun

1. Template:L

Further reading

• Template:R:gl:DRAG
• Template:R:gl:Estraviz

German

Etymology

Template:Bor+, from Template:Der. Template:Doublet.

Pronunciation

• Template:IPA
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Adjective

1. ideal Template:Gloss

Derived terms

• Template:L

Further reading

• Template:R:Duden
• Template:R:DWDS

Indonesian

Etymology

From Template:Bor, from Template:Der, from Template:Der.

Adjective

1. Template:L; optimal, being the best possibility
2. Template:L; pertaining to ideas, or to a given idea

Noun

1. Template:Lb Template:L; a subring closed under multiplication by its containing ring

Alternative forms

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Further reading

• Template:R:KBBI Daring

Luxembourgish

Pronunciation

• Template:IPA

Adjective

1. ideal

Norwegian Bokmål

Etymology

From Template:Der, from Template:Der, from Template:Der.

Noun

1. Template:L

References

• Template:R:The Bokmål Dictionary

Norwegian Nynorsk

Etymology

From Template:Der, from Template:Der, from Template:Der.

Noun

1. Template:L

References

• Template:R:The Nynorsk Dictionary

Piedmontese

Pronunciation

• Template:IPA

Adjective

1. Template:L

Noun

1. Template:L

Portuguese

Etymology

Template:Bor+.

Pronunciation

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• Template:Hyph

Adjective

1. Template:L
2. notional

Derived terms

• Template:L

Noun

1. Template:L
2. fantasy

Romanian

Etymology

Template:Bor+, from Template:Der.

Pronunciation

• Template:Audio

Adjective

1. Template:L

Related terms

• Template:L
• Template:L
• Template:L

Serbo-Croatian

Pronunciation

• Template:IPA
• Template:Hyph

Noun

1. Template:L

Spanish

Etymology

From Template:Der.

Adjective

1. Template:L

Noun

1. Template:L

Further reading

• Template:R:es:DRAE

Swedish

Pronunciation

• Template:Audio

Adjective

1. Template:L (perfect)

   Template:Syn

   Template:Ux

Noun

1. Template:L; perfect standard
2. Template:Lb Template:L; special subsets of a ring

Related terms

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References

• Template:R:svenska.se
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Anagrams

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Tagalog

Alternative forms

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Etymology

Template:Bor+.

Noun

1. Template:L Template:Gl

   Template:Syn

2. ambition

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Related terms

• Template:L
• Template:L
• Template:L

Adjective

1. Template:L Template:Gl
2. worthy of imitation

   Template:Syn

Turkish

Etymology

Borrowed from Template:Bor.

Pronunciation

• Template:IPA
• Template:Hyph

Adjective

1. ideal

Noun

1. ideal

Synonyms

• Template:L

Welsh

Alternative forms

• Template:Alt

Etymology

Template:Bor+.

Pronunciation

• Template:IPA

Adjective

1. Template:L

   Template:Syn

Noun

1. Template:L

   Template:Syn