ideal

== English == === 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. === Pronunciation === (UK) IPA(key): /aɪˈdɪəl/ (US) IPA(key): /aɪˈdi(ə)l/ (Indic) IPA(key): /ˈajɖɪjəl/ Rhymes: -iːəl === Adjective === ideal (comparative more ideal, superlative most ideal) Pertaining to ideas, or to a given idea. Existing only in the mind; conceptual, imaginary. Optimal; being the best possibility. 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; […] . Teaching or relating to the doctrine of idealism. (mathematics) Not actually present, but considered as present when limits at infinity are included. ==== Synonyms ==== (optimal): best, ideal, optimal, optimum (flawless): see also Thesaurus:flawless (of ideas): conceptual, notional (existing only in mind): conceptual, imaginary ==== Derived terms ==== ==== Related terms ==== ==== Translations ==== === Noun === ideal (plural ideals) 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 (algebra, ring theory) A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring. (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). 1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, 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. (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. (algebra, Lie theory) 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. (algebra) 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. ==== Synonyms ==== (order theory): order ideal (type of Lie subalgebra): Lie ideal ==== Antonyms ==== (antonym(s) of “order theory”): filter ==== Hyponyms ==== (mathematics): maximal ideal, principal ideal ==== Derived terms ==== ==== Translations ==== === References === === Further reading === Ideal (ring theory) on Wikipedia.Wikipedia Ideal (order theory) on Wikipedia.Wikipedia Ideal (set theory) on Wikipedia.Wikipedia Ideal point on Wikipedia.Wikipedia Ideal triangle on Wikipedia.Wikipedia Lie algebra on Wikipedia.Wikipedia === Anagrams === Delia, Elida, ailed, ladie == Asturian == === Etymology === From Latin ideālis. === Pronunciation === IPA(key): /ideˈal/ [i.ð̞eˈal] Rhymes: -al Syllabification: i‧de‧al === Adjective === ideal (epicene, plural ideales) ideal === Noun === ideal m (plural ideales) ideal === Further reading === “ideal (noun)”, in Diccionariu de la llingua asturiana [Dictionary of the Asturian Language] (in Asturian), 1ª edición, Academia de la Llingua Asturiana, 2000, →ISBN “ideal (adjective)”, in Diccionariu de la llingua asturiana [Dictionary of the Asturian Language] (in Asturian), 1ª edición, Academia de la Llingua Asturiana, 2000, →ISBN == Catalan == === Etymology === Borrowed from Latin ideālis. === Pronunciation === IPA(key): (Central, Balearic, Valencia) [i.ðeˈal] === Adjective === ideal m or f (masculine and feminine plural ideals) ideal ==== Derived terms ==== idealisme idealista idealitzar idealment === Noun === ideal m (plural ideals) ideal == Danish == === Etymology 1 === From Late Latin ideālis, from Latin idea. === Adjective === ideal (neuter idealt, plural and definite singular attributive ideale) ideal Synonyms: perfekt, fuldkommen Coordinate term: ideel === Inflection === ==== Derived terms ==== idealisere idealisme idealist === Etymology 2 === A nominalization of the adjective above. === Noun === ideal n (singular definite idealet, plural indefinite idealer) ideal (quality) role model, exemplar (person or thing) Synonym: forbillede ==== Declension ==== ==== Derived terms ==== skønhedsideal frihedsideal kvindeideal === Further reading === “ideal” in Den Danske Ordbog == Galician == === Etymology === From Latin ideālis. === Pronunciation === IPA(key): /idɛˈal/ [i.ð̞ɛˈɑɫ] Rhymes: -al Hyphenation: i‧de‧al === Adjective === ideal m or f (plural ideais) ideal ==== Derived terms ==== idealmente === Noun === ideal m (plural ideais) ideal === Further reading === “ideal”, in Dicionario da Real Academia Galega (in Galician), A Coruña: Royal Galician Academy, 2012–2026 “ideal”, in Dicionário Estraviz de galego (in Galician), 2014–2026 == German == === Etymology === Borrowed from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell. === Pronunciation === IPA(key): /ideˈaːl/ Rhymes: -aːl === Adjective === ideal (strong nominative masculine singular idealer, comparative (colloquial) idealer, superlative (colloquial) am idealsten) ideal (optimal, perfect) ==== Declension ==== ==== Derived terms ==== idealerweise === Further reading === “ideal” in Duden online “ideal” in Digitales Wörterbuch der deutschen Sprache == Indonesian == === Etymology === From Dutch ideaal, from French idéal, from Medieval Latin ideālis. === Pronunciation === (Standard Indonesian) IPA(key): /iˈdeal/ [iˈde.al] Rhymes: -eal Syllabification: i‧de‧al === Adjective === idéal (comparative lebih ideal, superlative paling ideal) ideal; optimal, being the best possibility ideal; pertaining to ideas, or to a given idea === Noun === idéal (plural ideal-ideal) (mathematics) ideal; a subring closed under multiplication by its containing ring ==== Alternative forms ==== idiil (noun) === Derived terms === === Related terms === === Further reading === “ideal”, in Kamus Besar Bahasa Indonesia [Great Dictionary of the Indonesian Language] (in Indonesian), Jakarta: Agency for Language Development and Cultivation – Ministry of Education, Culture, Research, and Technology of the Republic of Indonesia, 2016 == Luxembourgish == === Pronunciation === IPA(key): /ideˈaːl/, [idəˈaːl], /idiˈaːl/ === Adjective === ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten) ideal ==== Declension ==== == Norwegian Bokmål == === Etymology === From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). === Noun === ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene) ideal === References === “ideal” in The Bokmål Dictionary. == Norwegian Nynorsk == === Etymology === From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”). === Noun === ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala) ideal === References === “ideal” in The Nynorsk Dictionary. == Piedmontese == === Pronunciation === IPA(key): /ideˈal/ === Adjective === ideal ideal === Noun === ideal m ideal == Portuguese == === Etymology === Borrowed from Latin ideālis, equivalent to ideia +‎ -al. === Pronunciation === Rhymes: -al, -aw Hyphenation: i‧de‧al === Adjective === ideal m or f (plural ideais) ideal notional ==== Derived terms ==== idealmente === Noun === ideal m (plural ideais) ideal fantasy === Further reading === “ideal”, in Dicionário Priberam da Língua Portuguesa (in Portuguese), Lisbon: Priberam, 2008–2026 == Romanian == === Etymology === Borrowed from French idéal, from Latin idealis. === Pronunciation === === Adjective === ideal m or n (feminine singular ideală, masculine plural ideali, feminine/neuter plural ideale) ideal ==== Declension ==== ==== Related terms ==== idealism idealist idealiza == Serbo-Croatian == === Pronunciation === IPA(key): /iděaːl/ Hyphenation: i‧de‧al === Noun === idèāl m inan (Cyrillic spelling идѐа̄л) ideal ==== Declension ==== == Spanish == === Etymology === From Latin ideālis. === Pronunciation === IPA(key): /ideˈal/ [i.ð̞eˈal] Rhymes: -al Syllabification: i‧de‧al === Adjective === ideal m or f (masculine and feminine plural ideales) ideal ==== Derived terms ==== === Noun === ideal m (plural ideales) ideal === Further reading === “ideal”, in Diccionario de la lengua española [Dictionary of the Spanish Language] (in Spanish), online version 23.8.1, Royal Spanish Academy [Spanish: Real Academia Española], 15 December 2025 == Swedish == === Pronunciation === === Adjective === ideal (not comparable) ideal (perfect) Synonym: idealisk ==== Declension ==== === Noun === ideal n ideal; perfect standard (mathematics) ideal; special subsets of a ring ==== Declension ==== ==== Related terms ==== === References === ideal in Svensk ordbok (SO) ideal in Svenska Akademiens ordlista (SAOL) ideal in Svenska Akademiens ordbok (SAOB) === Anagrams === ilade == Tagalog == === Alternative forms === ideyal === Etymology === Borrowed from Spanish ideal. === Pronunciation === (Standard Tagalog) IPA(key): /ʔideˈal/ [ʔɪ.d̪ɛˈal] Rhymes: -al Syllabification: i‧de‧al === Noun === ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔) ideal (model to be imitated) Synonyms: huwaran, uliran ambition Synonyms: mithi, mithiin, pangarap, ambisyon ==== Related terms ==== idea idealismo idealista === Adjective === ideál (Baybayin spelling ᜁᜇᜒᜌᜎ᜔) ideal (just as what one wants) worthy of imitation Synonym: uliran == Turkish == === Etymology === Borrowed from French idéal. === Pronunciation === IPA(key): /ideˈɑl/ Hyphenation: i‧de‧al === Adjective === ideal ideal === Noun === ideal (definite accusative ideali, plural idealler) ideal ==== Declension ==== ==== Synonyms ==== ülkü == Welsh == === Alternative forms === ideol === Etymology === Borrowed from English ideal. === Pronunciation === IPA(key): /ɪˈdɛal/ === Adjective === ideal (feminine singular ideal, plural ideal, not comparable) ideal Synonym: delfrydol === Noun === ideal m (plural idealau) ideal Synonym: delfryd === Mutation === === References ===