View source for ideal

{{also|Ideal|ideał|ideál|idéal}}
==English==
{{wikipedia|Ideal}}

===Etymology===
From {{der|en|fr|idéal}}, from {{der|en|LL.|ideālis|t=existing in idea}}, {{surf|en|idea|-al|nocap=1}}, from {{der|en|la|idea|t=idea}}; see {{m|en|idea}}.
<br>In mathematics, the noun [[ring theory]] sense was first introduced by German mathematician {{w|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===
* {{rhymes|en|iːəl|s=3}}
* {{IPA|en|/aɪˈdiːl/|/aɪˈdɪəl/|/aɪˈdiː.əl/}}
* {{audio|en|en-us-ideal.ogg|a=US}}

===Adjective===
{{en-adj}}

# [[pertaining|Pertaining]] to [[idea]]s, or to a given idea.
# [[exist|Existing]] only in the [[mind]]; [[conceptual]], [[imaginary]].
#* {{quote-text|en|year=1796|author=Matthew Lewis|title=The Monk|page=256|publisher=Folio Society|year_published=1985
|passage=The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by '''ideal''' terrors —}}
#* {{RQ:Mary Shelley Frankenstein|chapter=[[s:Frankenstein, or the Modern Prometheus (Revised Edition, 1831)/Chapter 4|Chapter 4]]|text=Life and death appeared to me '''ideal''' bounds, which I should first break through, and pour a torrent of light into our dark world.}}
#* {{RQ:Dickens Pickwick Papers|passage=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'''.}}
# [[optimal|Optimal]]; being the [[best]] [[possibility]].
# [[perfect|Perfect]], [[flawless]], having no [[defect]]s.
#* '''1751''' April 13, {{w|Samuel Johnson}}, ''{{w|The Rambler}}'', Number 112, reprinted in 1825, ''The Works of Samuel Johnson, LL. D.'', Volume 1, Jones & Company, [https://books.google.com.au/books?id=WigPAAAAYAAJ&pg=PA194&dq=%22There+will+always+be+a+wide+interval+between+practical+and+ideal%22&hl=en&sa=X&ved=0ahUKEwiI2daW6e3XAhXoj1QKHTuXDUEQ6AEILDAB#v=onepage&q=%22There%20will%20always%20be%20a%20wide%20interval%20between%20practical%20and%20ideal%22&f=false page 194],
#*: There will always be a wide interval between practical and '''ideal''' excellence;{{...}}.
# Teaching or relating to the doctrine of [[idealism]].
#: {{ux|en|the '''ideal''' theory or philosophy}}
# {{lb|en|mathematics}} Not actually present, but considered as present when [[limit]]s at [[infinity]] are included.
#: {{ux|en|'''ideal''' point}}
#: {{ux|en|An '''ideal''' triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.}}

====Synonyms====
* {{sense|optimal}} {{l|en|best}}, {{l|en|ideal}}, {{l|en|optimal}}, {{l|en|optimum}}
* {{sense|flawless}} see also [[Thesaurus:flawless]]
* {{sense|of ideas}} {{l|en|conceptual}}, {{l|en|notional}}
* {{sense|existing only in mind}} {{l|en|conceptual}}, {{l|en|imaginary}}

====Derived terms====
{{der3|en|ideal gas| ideal mixture|ideal solution|non-ideal gas|ideal point|ideal triangle|ideal gas law|ideal number|ideal-seeking behavior|ideal-typical}}

====Related terms====
{{rel3|en|ideality|idealless|ideally|idealism|idealistic|idealize}}

====Translations====
{{trans-top|being optimal}}
* Afrikaans: {{t|af|voorbeeldig}}, {{t|af|ideaal}}
* Arabic: {{t|ar|مِثَالِيّ}}
* Asturian: {{t|ast|ideal}}
* Belarusian: {{t|be|ідэа́льны}}
* Bulgarian: {{t+|bg|идеа́лен}}
* Catalan: {{t+|ca|ideal}}
* Chinese:
*: Mandarin: {{t+|cmn|理想|tr=lǐxiǎng de|alt=理想的}}
* Czech: {{t+|cs|ideální}}
* Dutch: {{t+|nl|ideaal}}, {{t+|nl|optimaal}}
* Esperanto: {{t|eo|ideala}}
* Finnish: {{t+|fi|ihanteellinen}}, {{t+|fi|ideaalinen}}, {{t+|fi|ideaali|alt=ideaali-}}, {{t+|fi|optimaalinen}}
* French: {{t+|fr|idéal}}
* Galician: {{t|gl|ideal}}
* German: {{t+|de|ideal}}, {{t|de|bestmöglich}}, {{t+|de|optimal}}
* Hindi: {{t+|hi|आदर्श}}, {{t+|hi|श्रेष्ठ}}, {{t+|hi|आइड़ियल}}
* Hungarian: {{t+|hu|ideális}}
* Ido: {{t+|io|ideala}}
* Irish: {{t|ga|idéalach}}
* Italian: {{t+|it|ideale}}
* Japanese: {{t+|ja|理想的|tr=りそうてきな, risōteki na|alt=理想的な}}, {{t+|ja|理想|tr=りそうな, risō na|alt=理想な}}
* Korean: {{t+|ko|이상적(理想的)}}, {{t|ko|리상적(理想的)}} {{qualifier|North Korea}}
* Latvian: {{t|lv|ideāls}}
* Luxembourgish: {{t|lb|ideal}}
* Macedonian: {{t|mk|идеа́лен}}
* Malayalam: {{t+|ml|ആദർശം}}
* Norman: {{t|nrf|idéal}}
* Persian: {{t+|fa|آرمانی|tr=ârmâni}}, {{t|fa|ایده‌آل|tr=ide-âl}}
* Polish: {{t+|pl|idealny|m}}
* Portuguese: {{t+|pt|ideal}}
* Russian: {{t+|ru|идеа́льный}}
* Slovak: {{t|sk|ideálny}}
* Slovene: {{t|sl|idealen}}
* Spanish: {{t+|es|ideal}}
* Telugu: {{t+|te|ఆదర్శ}}
* Ukrainian: {{t|uk|ідеа́льний}}
{{trans-bottom}}

{{trans-top|being perfect}}
* Afrikaans: {{t|af|ideaal}}
* Asturian: {{t|ast|ideal}}
* Belarusian: {{t|be|ідэа́льны}}, {{t|be|даскана́лы}}
* Bulgarian: {{t+|bg|съвърше́н}}, {{t+|bg|идеа́лен}}
* Catalan: {{t+|ca|ideal}}
* Czech: {{t+|cs|ideální}}
* Dutch: {{t+|nl|ideaal}}, {{t+|nl|perfect}}, {{t+|nl|volmaakt}}
* Esperanto: {{t|eo|ideala}}
* Finnish: {{t+|fi|ihanteellinen}}, {{t+|fi|ideaalinen}}
* French: {{t+|fr|idéal}}, {{t+|fr|parfait}}
* Galician: {{t|gl|ideal|m|f}}
* German: {{t+|de|ideal}}, {{t+|de|vollendet}}, {{t+|de|einwandfrei}}, {{t+|de|tadellos}}, {{t+|de|perfekt}}
* Greek: {{t+|el|ιδανικός}}
* Hindi: {{t+|hi|आदर्श}}, {{t+|hi|आइड़ियल}}, {{t+|hi|श्रेष्ठ}}
* Hungarian: {{t+|hu|tökéletes}}, {{t+|hu|eszményi}}
* Ido: {{t+|io|ideala}}
* Irish: {{t|ga|idéalach}}
* Manx: {{t|gv|slanjeant}}
* Maori: {{t|mi|takarepakore}}, {{t|mi|paruhi}}
* Norman: {{t|nrf|idéal}}
* Persian: {{t+|fa|کامل|tr=kâmel}}, {{t|fa|فربود|tr=farbud}}, {{t+|fa|درست|tr=dorost}}
* Polish: {{t+|pl|idealny|m}}
* Portuguese: {{t+|pt|ideal}}
* Russian: {{t+|ru|идеа́льный}}, {{t+|ru|соверше́нный}}
* Slovene: {{t|sl|idealen}}
* Spanish: {{t+|es|ideal}}
* Telugu: {{t+|te|ఆదర్శ}}
* Ukrainian: {{t|uk|ідеа́льний}}, {{t|uk|доскона́лий}}
{{trans-bottom}}

{{trans-top|conceptual}}
* Afrikaans: {{t|af|ideëel}}, {{t|af|denkbeeldig}}
* Asturian: {{t|ast|ideal}}
* Bulgarian: {{t+|bg|въобража́ем}}, {{t+|bg|абстра́ктен}}
* Catalan: {{t+|ca|ideal}}
* Dutch: {{t+|nl|ideëel}}, {{t+|nl|conceptueel}}
* Esperanto: {{t|eo|ideala}}
* Finnish: {{t+|fi|ideaali|alt=ideaali-}}
* French: {{t+|fr|idéal}}
* Galician: {{t|gl|ideal}}
* German: {{t+|de|ideell}}
* Hebrew: {{t|he|אידיאלי|tr=ideàli}}
* Hungarian: {{t+|hu|képzelt}}
* Ido: {{t+|io|ideala}}
* Malayalam: {{t+|ml|ആദർശം}}
* Manx: {{t|gv|ard-smooinagh}}, {{t|gv|ard-smooinaghtagh}}
* Norman: {{t|nrf|idéal}}
* Polish: {{t+|pl|pojęciowy}}, {{t+|pl|konceptualny}}
* Portuguese: {{t+|pt|ideal}}
* Russian: {{t+|ru|идеа́льный}}, {{t+|ru|абстра́ктный}}, {{t+|ru|вообража́емый}}
* Slovene: {{t|sl|idejen}}
* Spanish: {{t+|es|ideal}}
{{trans-bottom}}

{{trans-top|at infinity}}
* Persian: {{t+|fa|غایی|tr=ğâyi}}
{{trans-bottom}}

{{checktrans-top}}
* Esperanto: {{t-check|eo|perfekta}}
* Indonesian: {{t+check|id|ideal}}
* Interlingua: {{t-check|ia|ideal}}
* Italian: {{t+check|it|ideale}}
* Spanish: {{t+check|es|ideal}}
{{trans-bottom}}

===Noun===
{{en-noun}}

# A thing which exists in the mind but not in reality; in ontological terms, a thing which has [[essence]] but not [[existence]].
# {{senseid|en|a perfect standard of beauty, intellect etc.}}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'' - [[w:Carl Schurz|Carl Schurz]]
#* {{quote-av
|en
|date=16 April 1945
|author=w:Harry S. Truman
|title=MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S.
|url=https://www.youtube.com/watch?v=Kil5e8GEEpU
|publisher=w:Harry S. Truman Presidential Library and Museum
|id=National Archives Identifier: [https://catalog.archives.gov/id/595162 595162]
|time=9:21
|text=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.}}
# {{lb|en|algebra|ring theory}} A [[subring]] closed under multiplication by its containing [[ring]].
#: {{ux|en|Let <math>\mathbb{Z}</math> be the ring of integers and let <math>2\mathbb{Z}</math> be its '''ideal''' of even integers. Then the quotient ring <math>\mathbb{Z} / 2\mathbb{Z}</math> is a Boolean ring.}}
#: {{ux|en|The product of two ideals <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is an ideal <math>\mathfrak{a b}</math> which is a subset of the intersection of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math>. This should help to understand why maximal '''ideals''' are prime '''ideals'''. Likewise, the union of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is a subset of <math>\mathfrak{a + b}</math>.}}
#* {{quote-book|en|year=2004|author=K. R. Goodearl; R. B. Warfield, Jr.|title=An Introduction to Noncommutative Noetherian Rings|edition=2nd|pageurl=https://books.google.com.au/books?id=vXh1MtvQr_8C&pg=PA47&dq=%22ideal%22%7C%22ideals%22+ring&hl=en&sa=X&ved=0ahUKEwjS54O5qdPdAhWL-VQKHVHvDWIQ6AEIngEwFQ#v=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=47|publisher=w:Cambridge University Press
|passage=In trying to understand the '''ideal''' theory of a commutative ring, one quickly sees that it is important to first understand the prime '''ideals'''.}}
#* {{quote-book|en|year=2009|author=John J. Watkins|title=Topics in Commutative Ring Theory|pageurl=https://books.google.com.au/books?id=Ln4u1bv5MT0C&pg=PA45&dq=%22ideal%22%7C%22ideals%22+ring&hl=en&sa=X&ved=0ahUKEwjS54O5qdPdAhWL-VQKHVHvDWIQ6AEIfDAP#v=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=45|publisher=w:Princeton University Press
|passage=If an '''ideal''' ''I'' of a ring contains the multiplicative identity 1, then we have seen that ''I'' must be the entire ring.}}
#* {{quote-book|en|year=2010|author=W. D. Burgess; A. Lashgari; A. Mojiri|chapter=Elements of Minimal Prime '''Ideals''' in General Rings|editors=Sergio R. López-Permouth; Dinh Van Huynh|title=Advances in Ring Theory|publisher=Springer (Birkhäuser)|pageurl=https://books.google.com.au/books?id=fCq_C8e8ZKYC&pg=PA69&dq=%22ideal%22%7C%22ideals%22+ring&hl=en&sa=X&ved=0ahUKEwjS54O5qdPdAhWL-VQKHVHvDWIQ6AEI7QEwIg#v=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=69
|passage=However, every ''R'' has a minimal prime '''ideal''' consisting of left zero-divisors and one of right zero-divisors.}}
# {{lb|en|algebra|order theory|lattice theory}} A non-[[empty set|empty]] [[lower set]] (of a [[partially ordered set]]) which is [[closure|closed]] under binary [[suprema]] (a.k.a. [[join]]s).<ref>{{pedia|Boolean prime ideal theorem#Prime ideal theorems}}</ref>
#* '''1992''', Unnamed translator, T. S. Fofanova, ''General Theory of Lattices'', in ''Ordered Sets and Lattices II'', {{w|American Mathematical Society}}, [https://books.google.com.au/books?id=WgWOh4G7StcC&pg=PA119&dq=%22ideal%22%7C%22ideals%22+lattice&hl=en&sa=X&ved=0ahUKEwiHlaLcrtXdAhXNAYgKHYOxCe0Q6AEIMzAC#v=onepage&q=%22ideal%22%7C%22ideals%22%20lattice&f=false 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.
#* {{quote-book|en|year=2011|author=George Grätzer|title=Lattice Theory: Foundation|publisher=Springer (Birkhäuser)|url=https://books.google.com.au/books?id=6XJX5-zCoIQC&pg=PA125&dq=%22ideal%22%7C%22ideals%22+lattice&hl=en&sa=X&ved=0ahUKEwiHlaLcrtXdAhXNAYgKHYOxCe0Q6AEILTAB#v=onepage&q=%22ideal%22%7C%22ideals%22%20lattice&f=false|page=125
|passage=1.35 Find a distributive lattice ''L'' with no minimal and no maximal prime '''ideals'''.}}
#* {{quote-book|en|year=2015|author=Vijay K. Garg|title=Introduction to Lattice Theory with Computer Science Applications|pageurl=https://books.google.com.au/books?id=eTlPCAAAQBAJ&pg=PA186&dq=%22ideal%22%7C%22ideals%22+lattice&hl=en&sa=X&ved=0ahUKEwiHlaLcrtXdAhXNAYgKHYOxCe0Q6AEIPjAE#v=onepage&q=%22ideal%22%7C%22ideals%22%20lattice&f=false|page=186|publisher=Wiley
|passage='''Definition 15.11''' ('''Width Ideal''') ''An '''ideal''' Q of a poset P = (X,≤) is a width '''ideal''' if maximal(Q) is a width antichain.''}}
# {{lb|en|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''' <math>I</math> of a given set <math>X</math> is a nonempty subset of the [[powerset]] <math>\mathcal{P}(X)</math> such that: <math>(1)\ \emptyset \in I</math>, <math>(2)\ A \in I \and B \subseteq A\implies B\in I</math> and <math>(3)\ A,B \in I\implies A\cup B \in I</math>.''
# {{lb|en|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.
#* {{quote-book|en|year=1975|translator=Che-Young Lee|author=Zhe-Xian Wan|title=Lie Algebras|publisher=Pergamon Press|pageurl=https://books.google.com.au/books?id=_4riBQAAQBAJ&pg=PA13&dq=%22ideal%22%7C%22ideals%22+lie&hl=en&sa=X&ved=0ahUKEwiF98-Q5dPdAhXQMt4KHQKvDPYQ6AEIRDAF#v=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=13
|passage=If 𝖌 is a Lie algebra, 𝖍 is an '''ideal''' and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.}}
#* {{quote-book|en|year=2006|author=W. McGovern|chapter=The work of Anthony Joseph in classical representation theory|editors=Anthony Joseph; Joseph Bernstein; Vladimir Hinich; Anna Melnikov|title=Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday|publisher=Springer (Birkhäuser)|pageurl=https://books.google.com.au/books?id=ODI4HVANcZMC&pg=PA3&dq=%22ideal%22%7C%22ideals%22+lie&hl=en&sa=X&ved=0ahUKEwiF98-Q5dPdAhXQMt4KHQKvDPYQ6AEISjAG#v=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=3
|passage=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.}}
#* {{quote-book|en|year=2013|author=J.E. Humphreys|title=Introduction to Lie Algebras and Representation Theory|pageurl=https://books.google.com.au/books?id=TeMlBQAAQBAJ&pg=PA73&dq=%22ideal%22%7C%22ideals%22+lie&hl=en&sa=X&ved=0ahUKEwiF98-Q5dPdAhXQMt4KHQKvDPYQ6AEIMzAC#v=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=73|publisher=Springer
|passage=Next let <math>L</math> be an arbitrary semisimple Lie algebra. Then <math>L</math> can be written uniquely as a direct sum <math>L_1\oplus \dots \oplus L_t</math> of simple '''ideals''' (Theorem 5.2).}}
# {{lb|en|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.<ref>{{cite-web|1=en|year=2004|author={{w|Vaughan Pratt}}|work=Chapter 1 : Lattice Theory|url=http://boole.stanford.edu/cs353/handouts/book1.pdf|publisher=boole.stanford.edu|section=&sect;1.3.5}}</ref>
#: {{ux|en|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====
* {{sense|order theory}} {{l|en|order ideal}}
* {{sense|type of Lie subalgebra}} {{l|en|Lie ideal}}

====Antonyms====
* {{antsense|order theory}} {{l|en|filter}}

====Hyponyms====
* {{sense|mathematics}} [[maximal ideal]], [[principal ideal]]

====Derived terms====
{{der4|en|ideal lattice|zero ideal|lattice ideal|left ideal|right ideal|two-sided ideal|Platonic ideal|prime ideal|principal ideal|pseudoideal|subideal|factor ideal|fractional ideal|Frink ideal|minimal ideal|primary ideal|principal ideal domain|principal ideal ring|radical ideal|vanishing ideal}}

====Translations====
{{trans-top|perfect standard of beauty, intellect etc.}}
* Afrikaans: {{t|af|ideaal}}
* Asturian: {{t|ast|ideal|m}}
* Belarusian: {{t|be|ідэа́л|m}}
* Bulgarian: {{t+|bg|идеа́л|m}}
* Catalan: {{t+|ca|ideal|m}}
* Chinese:
*: Mandarin: {{t+|cmn|典范|tr=diǎnfàn}}, {{t+|cmn|理想|tr=lǐxiǎng}}
* Czech: {{t|cs|ideál|m}}
* Dutch: {{t+|nl|ideaal|n}}, {{t+|nl|perfectie|f}}, {{t+|nl|streefdoel|n}}
* Esperanto: {{t|eo|ideala}}, {{t|eo|perfekta}}
* Finnish: {{t+|fi|ihanne}}, {{t+|fi|ideaali}}
* French: {{t+|fr|idéal|m}}
* Galician: {{t|gl|ideal|m}}
* German: {{t+|de|Ideal|n}}
* Greek: {{t+|el|ιδεώδες|n}}
* Hebrew: {{t+|he|אידיאל|tr=ide'ál|alt=אידיאל \ אִידֵאָל}}
* Hindi: {{t+|hi|आइड़ियल|m}}, {{t+|hi|आदर्श|m}}, {{t+|hi|श्रेष्ठ|m}}
* Hungarian: {{t+|hu|eszmény}}, {{t+|hu|ideál}}, {{t+|hu|mintakép}}
* Ido: {{t+|io|idealo}}
* Indonesian: {{t+|id|ideal}}
* Interlingua: {{t|ia|ideal}}
* Irish: {{t|ga|idéal|m}}
* Italian: {{t+|it|ideale|m}}
* Japanese: {{t+|ja|理想|tr=りそう, risō}}
* Kazakh: {{t|kk|мұрат}}
* Korean: {{t+|ko|이상(理想)}}, {{t+|ko|리상(理想)}} {{qualifier|North Korea}}
* Persian: {{t+|fa|آرمان|tr=ârmân}}, {{t|fa|ایدئال|tr=ide'âl}}
* Polish: {{t+|pl|ideał|m}}
* Portuguese: {{t+|pt|ideal|m}}
* Punjabi: {{t+|pa|ਆਦਰਸ਼|m}}
* Russian: {{t+|ru|идеа́л|m}}
* Slovak: {{t|sk|ideál|m}}
* Slovene: {{t|sl|ideal|m}}
* Spanish: {{t+|es|ideal|m}}
* Swedish: {{t+|sv|ideal|n}}
* Telugu: {{t+|te|ఆదర్శం}}
* Ukrainian: {{t|uk|ідеа́л|m}}
* Vietnamese: {{t+|vi|lý tưởng}}
* Welsh: {{t+|cy|delfryd|m|f}}
{{trans-bottom}}

{{trans-top|subring closed under multiplication by containing ring}}
* Chinese:
*: Mandarin: {{t+|cmn|理想|tr=lǐxiǎng}}
* Dutch: {{t+|nl|ideaal|n}}
* Finnish: {{t+|fi|ideaali}}
* French: {{t+|fr|idéal|m}}
* German: {{t+|de|Ideal|n}}
* Japanese: {{t|ja|イデアル|tr=idearu}}
* Kazakh: {{t|kk|идеал}}
* Polish: {{t+|pl|ideał|m}}
* Romanian: {{t+|ro|ideal|n}}
* Russian: {{t+|ru|идеа́л|m}}
* Swedish: {{t+|sv|ideal|n}}
{{trans-bottom}}

{{trans-top|lower set closed under joins}}
* Chinese:
*: Mandarin: {{t+|cmn|理想|tr=lǐxiǎng}}
* Irish: {{t|ga|idéal|m}}
* Russian: {{t+|ru|идеа́л|m}}
{{trans-bottom}}

{{trans-top|unattainable state}}
* German: {{t+|de|Idealzustand|m}}
* Polish: {{t+|pl|ideał|m}}
* Russian: {{t+|ru|идеа́л|m}}
{{trans-bottom}}

===References===
<references />

===Further reading===
* {{pedia|Ideal (ring theory)}}
* {{pedia|Ideal (order theory)}}
* {{pedia|Ideal (set theory)}}
* {{pedia|Ideal point}}
* {{pedia|Ideal triangle}}
* {{pedia|Lie algebra#Subalgebras, ideals and homomorphisms}}

===Anagrams===
* {{anagrams|en|a=adeil|Delia|Elida|ailed|ladie}}

{{C|en|Infinity|Non-Euclidean geometry}}

==Asturian==

===Etymology===
From {{der|ast|la|ideālis}}.

===Pronunciation===
{{ast-IPA}}

===Adjective===
{{ast-adj-mf|pl=ideales}}

# {{l|en|ideal}}

===Noun===
{{ast-noun|m|ideales}}

# {{l|en|ideal}}

==Catalan==

===Etymology===
{{bor+|ca|la|ideālis}}.

===Pronunciation===
* {{ca-IPA}}

===Adjective===
{{ca-adj}}

# {{l|en|ideal}}

====Derived terms====
* {{l|ca|idealisme}}
* {{l|ca|idealista}}
* {{l|ca|idealitzar}}
* {{l|ca|idealment}}

===Noun===
{{ca-noun|m}}

# {{l|en|ideal}}

==Galician==

===Etymology===
From {{der|gl|la|ideālis}}.

===Pronunciation===
{{gl-pr|idẹal}}
* {{hyph|gl|i|de|al}}

===Adjective===
{{gl-adj}}

# {{l|en|ideal}}

====Derived terms====
* {{l|gl|idealmente}}

===Noun===
{{gl-noun|m}}

# {{l|en|ideal}}

===Further reading===
* {{R:gl:DRAG}}
* {{R:gl:Estraviz}}

==German==

===Etymology===
{{bor+|de|LL.|ideālis|t=existing in idea}}, from {{der|de|la|idea|t=idea}}. {{doublet|de|ideell}}.

===Pronunciation===
* {{IPA|de|/ideˈaːl/}}
* {{rhymes|de|aːl|s=3}}
* {{audio|de|De-ideal.ogg}}

===Adjective===
{{de-adj|comp:+[colloquial]}}

# [[#English|ideal]] {{gloss|optimal, perfect}}

====Declension====
{{de-adecl|comp}}

====Derived terms====
* {{l|de|idealerweise}}

===Further reading===
* {{R:Duden}}
* {{R:DWDS}}

==Indonesian==
{{wikipedia||lang=id}}

===Etymology===
From {{bor|id|nl|ideaal}}, from {{der|id|fr|idéal}}, from {{der|id|ML.|ideālis}}.

===Pronunciation===
* {{IPA|id|[ˈɪdeal]}}
* {{hyphenation|id|idé|al}}

===Adjective===
{{id-adj|head=|pl=-}}

# {{l|en|ideal}}:
## optimal; being the best possibility.
## pertaining to ideas, or to a given idea.

===Noun===
{{id-noun|head=|pl=ideal-ideal}}

# {{lb|id|mathematics}} {{l|en|ideal}}: a subring closed under multiplication by its containing ring.

====Alternative forms====
* {{alter|id|idiil}}

===Affixed terms===
{{der4|id|
| mengidealkan
}}

===Related terms===
{{rel4|id|
| ide
}}

===Further reading===
* {{R:KBBI Daring}}

==Luxembourgish==

===Pronunciation===
* {{IPA|lb|/ideˈaːl/|[idəˈaːl]|/idiˈaːl/}}

===Adjective===
{{lb-adj||ideaalt|sup=ideaalsten}}

# [[ideal#English|ideal]]

====Declension====
{{lb-decl-adj|long=ideaal}}

==Norwegian Bokmål==

===Etymology===
From {{der|nb|fr|idéal}}, from {{der|nb|LL.|ideālis|t=existing in idea}}, from {{der|nb|la|idea|t=idea}}.

===Noun===
{{nb-noun-n3}}

# {{l|en|ideal}}

===References===
* {{R:The Bokmål Dictionary}}

==Norwegian Nynorsk==

===Etymology===
From {{der|nn|fr|idéal}}, from {{der|nn|LL.|ideālis|t=existing in idea}}, from {{der|nn|la|idea|t=idea}}.

===Noun===
{{nn-noun-n1}}

# {{l|en|ideal}}

===References===
* {{R:The Nynorsk Dictionary}}

==Piedmontese==

===Pronunciation===
* {{IPA|pms|/ideˈal/}}

===Adjective===
{{head|pms|adjective}}

# {{l|en|ideal}}

===Noun===
{{head|pms|noun|g=m}}

# {{l|en|ideal}}

==Portuguese==

===Etymology===
{{bor+|pt|la|ideālis}}.

===Pronunciation===
{{pt-IPA}}
* {{rhymes|pt|al|aw|s=3}}
* {{hyph|pt|i|de|al}}

===Adjective===
{{pt-adj}}

# {{l|en|ideal}}
# [[notional]]

====Derived terms====
* {{l|pt|idealmente}}

===Noun===
{{pt-noun|m}}

# {{l|en|ideal}}
# [[fantasy]]

==Romanian==

===Etymology===
{{bor+|ro|fr|idéal}}, from {{der|ro|la|idealis}}.

===Pronunciation===
* {{audio|ro|LL-Q7913 (ron)-Andreea Teodoraa-ideal.wav}}

===Adjective===
{{ro-adj}}

# {{l|en|ideal}}

====Declension====
{{ro-decl-adj}}

====Related terms====
* {{l|ro|idealism}}
* {{l|ro|idealist}}
* {{l|ro|idealiza}}

==Serbo-Croatian==

===Pronunciation===
* {{IPA|sh|/iděaːl/}}
* {{hyph|sh|i|de|al}}

===Noun===
{{sh-noun|g=m|head=idèāl}}

# {{l|en|ideal}}

====Declension====
{{sh-decl-noun
|idèāl|ideali
|ideála|ideala
|idealu|idealima
|ideal|ideale
|ideale|ideali
|idealu|idealima
|idealom|idealima
}}

==Spanish==

===Etymology===
From {{der|es|la|ideālis}}.

===Pronunciation===
{{es-pr}}

===Adjective===
{{es-adj}}

# {{l|en|ideal}}

====Derived terms====
{{col-auto|es|idealidad|idealizar|idealmente}}

===Noun===
{{es-noun|m}}

# {{l|en|ideal}}

===Further reading===
* {{R:es:DRAE}}

==Swedish==

===Pronunciation===
* {{audio|sv|Sv-ideal.ogg}}

===Adjective===
{{sv-adj|-}}

# {{l|en|ideal}} (perfect)
#: {{syn|sv|idealisk}}
#: {{ux|sv|under '''ideala''' förhållanden|under '''ideal''' conditions}}

====Declension====
{{sv-adj-abs}}

===Noun===
{{sv-noun|n}}

# {{l|en|ideal}}; perfect standard
# {{lb|sv|mathematics}} {{l|en|ideal}}; special subsets of a [[ring]]

====Declension====
{{sv-infl-noun-n-zero}}

====Related terms====
* {{l|sv|idé}}
* {{l|sv|idealisera}}
* {{l|sv|idealisk}}
* {{l|sv|idealism}}
* {{l|sv|idealist}}
* {{l|sv|idealistisk}}

===References===
* {{R:svenska.se|so}}
* {{R:svenska.se|saol}}
* {{R:svenska.se|saob}}

===Anagrams===
* {{anagrams|sv|a=adeil|ilade}}

==Tagalog==

===Alternative forms===
* {{alt|tl|ideyal}}

===Etymology===
{{bor+|tl|es|ideal}}.

===Pronunciation===
{{tl-pr|ideyál}}

===Noun===
{{tl-noun|ideál|b=ideyal}}

# {{l|en|ideal}} {{gl|model to be imitated}}
#: {{syn|tl|huwaran|uliran}}
# [[ambition]]
#: {{syn|tl|mithi|mithiin|pangarap|ambisyon}}

====Related terms====
* {{l|tl|idea}}
* {{l|tl|idealismo}}
* {{l|tl|idealista}}

===Adjective===
{{tl-adj|ideál|b=ideyal}}

# {{l|en|ideal}} {{gl|just as what one wants}}
# [[worthy]] of [[imitation]]
#: {{syn|tl|uliran}}

==Turkish==

===Etymology===
Borrowed from {{bor|tr|fr|idéal}}.

===Pronunciation===
* {{IPA|tr|/ideˈɑl/}}
* {{hyph|tr|i|de|al}}

===Adjective===
{{tr-adj}}

# [[#English|ideal]]

===Noun===
{{tr-noun|i|ler}}

# [[#English|ideal]]

====Declension====
{{tr-infl-noun-c|e|poss=1|pred=1}}

====Synonyms====
* {{l|tr|ülkü}}

==Welsh==

===Alternative forms===
* {{alt|cy|ideol}}

===Etymology===
{{bor+|cy|en|ideal}}.<ref name=GPC>{{R:cy:GPC}}</ref>

===Pronunciation===
* {{IPA|cy|/ɪˈdɛal/}}

===Adjective===
{{cy-adj|-}}

# {{l|en|ideal}}<ref name=GPC></ref><ref name=Termiadur>{{R:cy:Termiadur}}</ref>
#: {{syn|cy|delfrydol}}

===Noun===
{{cy-noun|m|au}}

# {{l|en|ideal}}<ref name=GPC></ref><ref name=Termiadur></ref>
#: {{syn|cy|delfryd}}

===Mutation===
{{cy-mut}}

===References===
{{reflist}}