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ideal/English/noun: invalid uppercase tag North-Korea not in or uppercase_tags: {"antonyms": [{"sense": "antonym(s) of “order theory”", "word": "filter"}], "categories": ["English adjectives", "English countable nouns", "English entries with incorrect language header", "English lemmas", "English nouns", "English terms derived from French", "English terms derived from Late Latin", "English terms derived from Latin", "English terms suffixed with -al", "Entries with translation boxes", "Pages with 18 entries", "Pages with entries", "Requests for review of Esperanto translations", "Requests for review of Indonesian translations", "Requests for review of Interlingua translations", "Requests for review of Italian translations", "Rhymes:English/iːəl", "Rhymes:English/iːəl/3 syllables", "Terms with Afrikaans translations", "Terms with Arabic translations", "Terms with Armenian translations", "Terms with Asturian translations", "Terms with Belarusian translations", "Terms with Bulgarian translations", "Terms with Catalan translations", "Terms with Czech translations", "Terms with Dutch translations", "Terms with Esperanto translations", "Terms with Finnish translations", "Terms with French translations", "Terms with Galician translations", "Terms with German translations", "Terms with Greek translations", "Terms with Hebrew translations", "Terms with Hindi translations", "Terms with Hungarian translations", "Terms with Ido translations", "Terms with Indonesian translations", "Terms with Interlingua translations", "Terms with Irish translations", "Terms with Italian translations", "Terms with Japanese translations", "Terms with Kazakh translations", "Terms with Korean translations", "Terms with Latvian translations", "Terms with Luxembourgish translations", "Terms with Macedonian translations", "Terms with Malayalam translations", "Terms with Mandarin translations", "Terms with Manx translations", "Terms with Maori translations", "Terms with Norman translations", "Terms with Pannonian Rusyn translations", "Terms with Persian translations", "Terms with Polish translations", "Terms with Portuguese translations", "Terms with Punjabi translations", "Terms with Romanian translations", "Terms with Russian translations", "Terms with Slovak translations", "Terms with Slovene translations", "Terms with Spanish translations", "Terms with Swedish translations", "Terms with Telugu translations", "Terms with Ukrainian translations", "Terms with Vietnamese translations", "Terms with Welsh translations", "en:Infinity", "en:Non-Euclidean geometry"], "derived": [{"word": "factor ideal"}, {"word": "fractional ideal"}, {"word": "Frink ideal"}, {"word": "ideal lattice"}, {"word": "lattice ideal"}, {"word": "left ideal"}, {"word": "minimal ideal"}, {"word": "Platonic ideal"}, {"word": "primary ideal"}, {"word": "prime ideal"}, {"word": "principal ideal"}, {"word": "principal ideal domain"}, {"word": "principal ideal ring"}, {"word": "pseudoideal"}, {"word": "radical ideal"}, {"word": "right ideal"}, {"word": "subideal"}, {"word": "two-sided ideal"}, {"word": "vanishing ideal"}, {"word": "zero ideal"}], "etymology_templates": [{"args": {"1": "en", "2": "fr", "3": "idéal"}, "expansion": "French idéal", "name": "der"}, {"args": {"1": "en", "2": "LL.", "3": "ideālis", "t": "existing in idea"}, "expansion": "Late Latin ideālis (“existing in idea”)", "name": "der"}, {"args": {"1": "en", "2": "idea", "3": "-al", "nocap": "1"}, "expansion": "by surface analysis, idea + -al", "name": "surf"}, {"args": {"1": "en", "2": "la", "3": "idea", "t": "idea"}, "expansion": "Latin idea (“idea”)", "name": "der"}], "etymology_text": "From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.\nIn 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.", "forms": [{"form": "ideals", "tags": ["plural"]}], "head_templates": [{"args": {}, "expansion": "ideal (plural ideals)", "name": "en-noun"}], "hyponyms": [{"topics": ["mathematics", "sciences"], "word": "maximal ideal"}, {"topics": ["mathematics", "sciences"], "word": "principal ideal"}], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [{"glosses": ["A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence."], "links": [["essence", "essence"], ["existence", "existence"]]}, {"categories": ["English terms with quotations"], "examples": [{"text": "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"}, {"ref": "1945 April 16, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S., Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162:", "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.", "type": "quote"}], "glosses": ["A perfect standard of beauty, intellect etc., or a standard of excellence to aim at."], "links": [["perfect", "perfect"], ["standard", "standard"], ["excellence", "excellence"], ["aim", "aim"]], "senseid": ["en:a perfect standard of beauty, intellect etc."]}, {"categories": ["English terms with quotations", "English terms with usage examples", "en:Algebra"], "examples": [{"text": "Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.", "type": "example"}, {"text": "The product of two ideals #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is an ideal #92;mathfrak#123;ab#125; which is a subset of the intersection of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125;. This should help to understand why maximal ideals are prime ideals. Likewise, the union of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is a subset of #92;mathfrak#123;a#43;b#125;.", "type": "example"}, {"ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:", "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.", "type": "quote"}, {"ref": "2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45:", "text": "If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.", "type": "quote"}, {"ref": "2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69:", "text": "However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.", "type": "quote"}], "glosses": ["A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring."], "links": [["algebra", "algebra"], ["two-sided ideal", "two-sided ideal"], ["subset", "subset"], ["ring", "ring"], ["closed", "closed"], ["left", "left"], ["right", "right"], ["multiplication", "multiplication"], ["elements", "elements"]], "qualifier": "ring theory", "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}, {"categories": ["English terms with quotations", "en:Algebra"], "examples": [{"text": "1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,\nAn 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."}, {"ref": "2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125:", "text": "1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.", "type": "quote"}, {"ref": "2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186:", "text": "Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.", "type": "quote"}], "glosses": ["A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins)."], "links": [["algebra", "algebra"], ["empty", "empty set"], ["lower set", "lower set"], ["partially ordered set", "partially ordered set"], ["closed", "closure"], ["suprema", "suprema"], ["join", "join"], ["Boolean prime ideal theorem", "w:Boolean prime ideal theorem#Prime ideal theorems"]], "qualifier": "lattice theory", "raw_glosses": ["(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)."], "topics": ["algebra", "mathematics", "order-theory", "sciences"]}, {"categories": ["English terms with usage examples", "en:Set theory"], "examples": [{"text": "Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.", "type": "example"}], "glosses": ["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."], "links": [["set theory", "set theory"], ["small", "small"], ["negligible", "negligible"], ["subset", "subset"], ["union", "union"]], "raw_glosses": ["(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."], "topics": ["mathematics", "sciences", "set-theory"]}, {"categories": ["English terms with quotations", "en:Algebra"], "examples": [{"ref": "1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13:", "text": "If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.", "type": "quote"}, {"ref": "2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov, editors, Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3:", "text": "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.", "type": "quote"}, {"ref": "2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73:", "text": "Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).", "type": "quote"}], "glosses": ["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."], "links": [["algebra", "algebra"], ["Lie subalgebra", "Lie subalgebra"], ["Lie bracket", "Lie bracket"], ["Lie algebra", "Lie algebra"], ["subset", "subset"]], "qualifier": "Lie theory", "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}, {"categories": ["English terms with usage examples", "en:Algebra"], "examples": [{"text": "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.", "type": "example"}], "glosses": ["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."], "links": [["algebra", "algebra"], ["subsemigroup", "subsemigroup"], ["semigroup", "semigroup"], ["Vaughan Pratt", "w:Vaughan Pratt"]], "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}], "sounds": [{"rhymes": "-iːəl"}, {"ipa": "/aɪˈdiːl/"}, {"ipa": "/aɪˈdɪəl/"}, {"ipa": "/aɪˈdiː.əl/"}, {"audio": "en-us-ideal.ogg", "mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/8/82/En-us-ideal.ogg/En-us-ideal.ogg.mp3", "ogg_url": "https://upload.wikimedia.org/wikipedia/commons/8/82/En-us-ideal.ogg"}], "synonyms": [{"topics": ["order-theory", "mathematics", "sciences"], "word": "order ideal"}, {"sense": "type of Lie subalgebra", "word": "Lie ideal"}], "translations": [{"code": "af", "lang": "Afrikaans", "sense": "perfect standard of beauty, intellect etc.", "word": "ideaal"}, {"code": "ast", "lang": "Asturian", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "be", "lang": "Belarusian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ідэа́л"}, {"code": "bg", "lang": "Bulgarian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "идеа́л"}, {"code": "ca", "lang": "Catalan", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "diǎnfàn", "sense": "perfect standard of beauty, intellect etc.", "word": "典范"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "perfect standard of beauty, intellect etc.", "word": "理想"}, {"code": "cs", "lang": "Czech", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideál"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ideaal"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["feminine"], "word": "perfectie"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "streefdoel"}, {"code": "eo", "lang": "Esperanto", "sense": "perfect standard of beauty, intellect etc.", "word": "idealo"}, {"code": "eo", "lang": "Esperanto", "sense": "perfect standard of beauty, intellect etc.", "word": "perfekta"}, {"code": "fi", "lang": "Finnish", "sense": "perfect standard of beauty, intellect etc.", "word": "ihanne"}, {"code": "fi", "lang": "Finnish", "sense": "perfect standard of beauty, intellect etc.", "word": "ideaali"}, {"code": "fr", "lang": "French", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "idéal"}, {"code": "gl", "lang": "Galician", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "de", "lang": "German", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "Ideal"}, {"code": "el", "lang": "Greek", "roman": "ideódes", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ιδεώδες"}, {"code": "he", "lang": "Hebrew", "roman": "ide'ál", "sense": "perfect standard of beauty, intellect etc.", "word": "אידיאל \\ אִידֵאָל"}, {"code": "hi", "lang": "Hindi", "roman": "āiṛiyal", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "आइड़ियल"}, {"code": "hi", "lang": "Hindi", "roman": "ādarś", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "आदर्श"}, {"code": "hi", "lang": "Hindi", "roman": "śreṣṭh", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "श्रेष्ठ"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "eszmény"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "ideál"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "mintakép"}, {"code": "io", "lang": "Ido", "sense": "perfect standard of beauty, intellect etc.", "word": "idealo"}, {"code": "id", "lang": "Indonesian", "sense": "perfect standard of beauty, intellect etc.", "word": "ideal"}, {"code": "ia", "lang": "Interlingua", "sense": "perfect standard of beauty, intellect etc.", "word": "ideal"}, {"code": "ga", "lang": "Irish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "idéal"}, {"code": "it", "lang": "Italian", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideale"}, {"alt": "りそう", "code": "ja", "lang": "Japanese", "roman": "risō", "sense": "perfect standard of beauty, intellect etc.", "word": "理想"}, {"code": "kk", "lang": "Kazakh", "roman": "mūrat", "sense": "perfect standard of beauty, intellect etc.", "word": "мұрат"}, {"alt": "理想", "code": "ko", "lang": "Korean", "roman": "isang", "sense": "perfect standard of beauty, intellect etc.", "word": "이상"}, {"alt": "理想", "code": "ko", "lang": "Korean", "roman": "risang", "sense": "perfect standard of beauty, intellect etc.", "tags": ["North-Korea"], "word": "리상"}, {"code": "fa", "lang": "Persian", "roman": "ârmân", "sense": "perfect standard of beauty, intellect etc.", "word": "آرمان"}, {"code": "fa", "lang": "Persian", "roman": "ide'âl", "sense": "perfect standard of beauty, intellect etc.", "word": "ایدئال"}, {"code": "pl", "lang": "Polish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideał"}, {"code": "pt", "lang": "Portuguese", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "pa", "lang": "Punjabi", "roman": "ādarś", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ਆਦਰਸ਼"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "идеа́л"}, {"code": "sk", "lang": "Slovak", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideál"}, {"code": "sl", "lang": "Slovene", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "es", "lang": "Spanish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "sv", "lang": "Swedish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ideal"}, {"code": "te", "lang": "Telugu", "roman": "ādarśaṁ", "sense": "perfect standard of beauty, intellect etc.", "word": "ఆదర్శం"}, {"code": "uk", "lang": "Ukrainian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ідеа́л"}, {"code": "vi", "lang": "Vietnamese", "sense": "perfect standard of beauty, intellect etc.", "word": "lý tưởng"}, {"code": "cy", "lang": "Welsh", "sense": "perfect standard of beauty, intellect etc.", "tags": ["feminine", "masculine"], "word": "delfryd"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "subring closed under multiplication by containing ring", "word": "理想"}, {"code": "nl", "lang": "Dutch", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideaal"}, {"code": "fi", "lang": "Finnish", "sense": "subring closed under multiplication by containing ring", "word": "ideaali"}, {"code": "fr", "lang": "French", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "idéal"}, {"code": "de", "lang": "German", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "Ideal"}, {"code": "ja", "lang": "Japanese", "roman": "idearu", "sense": "subring closed under multiplication by containing ring", "word": "イデアル"}, {"code": "kk", "lang": "Kazakh", "roman": "ideal", "sense": "subring closed under multiplication by containing ring", "word": "идеал"}, {"code": "pl", "lang": "Polish", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "ideał"}, {"code": "ro", "lang": "Romanian", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideal"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "идеа́л"}, {"code": "sv", "lang": "Swedish", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideal"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "lower set closed under joins", "word": "理想"}, {"code": "ga", "lang": "Irish", "sense": "lower set closed under joins", "tags": ["masculine"], "word": "idéal"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "lower set closed under joins", "tags": ["masculine"], "word": "идеа́л"}, {"code": "de", "lang": "German", "sense": "unattainable state", "tags": ["masculine"], "word": "Idealzustand"}, {"code": "pl", "lang": "Polish", "sense": "unattainable state", "tags": ["masculine"], "word": "ideał"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "unattainable state", "tags": ["masculine"], "word": "идеа́л"}], "wikipedia": ["Ideal", "Richard Dedekind"], "word": "ideal"}

ideal (English noun) ideal/English/noun: invalid uppercase tag North-Korea not in or uppercase_tags: {"antonyms": [{"sense": "antonym(s) of “order theory”", "word": "filter"}], "categories": ["English adjectives", "English countable nouns", "English entries with incorrect language header", "English lemmas", "English nouns", "English terms derived from French", "English terms derived from Late Latin", "English terms derived from Latin", "English terms suffixed with -al", "Entries with translation boxes", "Pages with 18 entries", "Pages with entries", "Requests for review of Esperanto translations", "Requests for review of Indonesian translations", "Requests for review of Interlingua translations", "Requests for review of Italian translations", "Rhymes:English/iːəl", "Rhymes:English/iːəl/3 syllables", "Terms with Afrikaans translations", "Terms with Arabic translations", "Terms with Armenian translations", "Terms with Asturian translations", "Terms with Belarusian translations", "Terms with Bulgarian translations", "Terms with Catalan translations", "Terms with Czech translations", "Terms with Dutch translations", "Terms with Esperanto translations", "Terms with Finnish translations", "Terms with French translations", "Terms with Galician translations", "Terms with German translations", "Terms with Greek translations", "Terms with Hebrew translations", "Terms with Hindi translations", "Terms with Hungarian translations", "Terms with Ido translations", "Terms with Indonesian translations", "Terms with Interlingua translations", "Terms with Irish translations", "Terms with Italian translations", "Terms with Japanese translations", "Terms with Kazakh translations", "Terms with Korean translations", "Terms with Latvian translations", "Terms with Luxembourgish translations", "Terms with Macedonian translations", "Terms with Malayalam translations", "Terms with Mandarin translations", "Terms with Manx translations", "Terms with Maori translations", "Terms with Norman translations", "Terms with Pannonian Rusyn translations", "Terms with Persian translations", "Terms with Polish translations", "Terms with Portuguese translations", "Terms with Punjabi translations", "Terms with Romanian translations", "Terms with Russian translations", "Terms with Slovak translations", "Terms with Slovene translations", "Terms with Spanish translations", "Terms with Swedish translations", "Terms with Telugu translations", "Terms with Ukrainian translations", "Terms with Vietnamese translations", "Terms with Welsh translations", "en:Infinity", "en:Non-Euclidean geometry"], "derived": [{"word": "factor ideal"}, {"word": "fractional ideal"}, {"word": "Frink ideal"}, {"word": "ideal lattice"}, {"word": "lattice ideal"}, {"word": "left ideal"}, {"word": "minimal ideal"}, {"word": "Platonic ideal"}, {"word": "primary ideal"}, {"word": "prime ideal"}, {"word": "principal ideal"}, {"word": "principal ideal domain"}, {"word": "principal ideal ring"}, {"word": "pseudoideal"}, {"word": "radical ideal"}, {"word": "right ideal"}, {"word": "subideal"}, {"word": "two-sided ideal"}, {"word": "vanishing ideal"}, {"word": "zero ideal"}], "etymology_templates": [{"args": {"1": "en", "2": "fr", "3": "idéal"}, "expansion": "French idéal", "name": "der"}, {"args": {"1": "en", "2": "LL.", "3": "ideālis", "t": "existing in idea"}, "expansion": "Late Latin ideālis (“existing in idea”)", "name": "der"}, {"args": {"1": "en", "2": "idea", "3": "-al", "nocap": "1"}, "expansion": "by surface analysis, idea + -al", "name": "surf"}, {"args": {"1": "en", "2": "la", "3": "idea", "t": "idea"}, "expansion": "Latin idea (“idea”)", "name": "der"}], "etymology_text": "From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.\nIn 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.", "forms": [{"form": "ideals", "tags": ["plural"]}], "head_templates": [{"args": {}, "expansion": "ideal (plural ideals)", "name": "en-noun"}], "hyponyms": [{"topics": ["mathematics", "sciences"], "word": "maximal ideal"}, {"topics": ["mathematics", "sciences"], "word": "principal ideal"}], "lang": "English", "lang_code": "en", "pos": "noun", "senses": [{"glosses": ["A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence."], "links": [["essence", "essence"], ["existence", "existence"]]}, {"categories": ["English terms with quotations"], "examples": [{"text": "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"}, {"ref": "1945 April 16, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S., Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162:", "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.", "type": "quote"}], "glosses": ["A perfect standard of beauty, intellect etc., or a standard of excellence to aim at."], "links": [["perfect", "perfect"], ["standard", "standard"], ["excellence", "excellence"], ["aim", "aim"]], "senseid": ["en:a perfect standard of beauty, intellect etc."]}, {"categories": ["English terms with quotations", "English terms with usage examples", "en:Algebra"], "examples": [{"text": "Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.", "type": "example"}, {"text": "The product of two ideals #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is an ideal #92;mathfrak#123;ab#125; which is a subset of the intersection of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125;. This should help to understand why maximal ideals are prime ideals. Likewise, the union of #92;mathfrak#123;a#125; and #92;mathfrak#123;b#125; is a subset of #92;mathfrak#123;a#43;b#125;.", "type": "example"}, {"ref": "2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47:", "text": "In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.", "type": "quote"}, {"ref": "2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45:", "text": "If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.", "type": "quote"}, {"ref": "2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69:", "text": "However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.", "type": "quote"}], "glosses": ["A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring."], "links": [["algebra", "algebra"], ["two-sided ideal", "two-sided ideal"], ["subset", "subset"], ["ring", "ring"], ["closed", "closed"], ["left", "left"], ["right", "right"], ["multiplication", "multiplication"], ["elements", "elements"]], "qualifier": "ring theory", "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}, {"categories": ["English terms with quotations", "en:Algebra"], "examples": [{"text": "1992, Unnamed translator, T. S. Fofanova, General Theory of Lattices, in Ordered Sets and Lattices II, American Mathematical Society, page 119,\nAn 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."}, {"ref": "2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125:", "text": "1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.", "type": "quote"}, {"ref": "2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186:", "text": "Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.", "type": "quote"}], "glosses": ["A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins)."], "links": [["algebra", "algebra"], ["empty", "empty set"], ["lower set", "lower set"], ["partially ordered set", "partially ordered set"], ["closed", "closure"], ["suprema", "suprema"], ["join", "join"], ["Boolean prime ideal theorem", "w:Boolean prime ideal theorem#Prime ideal theorems"]], "qualifier": "lattice theory", "raw_glosses": ["(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)."], "topics": ["algebra", "mathematics", "order-theory", "sciences"]}, {"categories": ["English terms with usage examples", "en:Set theory"], "examples": [{"text": "Formally, an ideal I of a given set X is a nonempty subset of the powerset #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.", "type": "example"}], "glosses": ["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."], "links": [["set theory", "set theory"], ["small", "small"], ["negligible", "negligible"], ["subset", "subset"], ["union", "union"]], "raw_glosses": ["(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."], "topics": ["mathematics", "sciences", "set-theory"]}, {"categories": ["English terms with quotations", "en:Algebra"], "examples": [{"ref": "1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13:", "text": "If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.", "type": "quote"}, {"ref": "2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov, editors, Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday, Springer (Birkhäuser), page 3:", "text": "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.", "type": "quote"}, {"ref": "2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73:", "text": "Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L#95;1#92;oplus#92;dots#92;oplusL#95;t of simple ideals (Theorem 5.2).", "type": "quote"}], "glosses": ["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."], "links": [["algebra", "algebra"], ["Lie subalgebra", "Lie subalgebra"], ["Lie bracket", "Lie bracket"], ["Lie algebra", "Lie algebra"], ["subset", "subset"]], "qualifier": "Lie theory", "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}, {"categories": ["English terms with usage examples", "en:Algebra"], "examples": [{"text": "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.", "type": "example"}], "glosses": ["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."], "links": [["algebra", "algebra"], ["subsemigroup", "subsemigroup"], ["semigroup", "semigroup"], ["Vaughan Pratt", "w:Vaughan Pratt"]], "raw_glosses": ["(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."], "topics": ["algebra", "mathematics", "sciences"]}], "sounds": [{"rhymes": "-iːəl"}, {"ipa": "/aɪˈdiːl/"}, {"ipa": "/aɪˈdɪəl/"}, {"ipa": "/aɪˈdiː.əl/"}, {"audio": "en-us-ideal.ogg", "mp3_url": "https://upload.wikimedia.org/wikipedia/commons/transcoded/8/82/En-us-ideal.ogg/En-us-ideal.ogg.mp3", "ogg_url": "https://upload.wikimedia.org/wikipedia/commons/8/82/En-us-ideal.ogg"}], "synonyms": [{"topics": ["order-theory", "mathematics", "sciences"], "word": "order ideal"}, {"sense": "type of Lie subalgebra", "word": "Lie ideal"}], "translations": [{"code": "af", "lang": "Afrikaans", "sense": "perfect standard of beauty, intellect etc.", "word": "ideaal"}, {"code": "ast", "lang": "Asturian", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "be", "lang": "Belarusian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ідэа́л"}, {"code": "bg", "lang": "Bulgarian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "идеа́л"}, {"code": "ca", "lang": "Catalan", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "diǎnfàn", "sense": "perfect standard of beauty, intellect etc.", "word": "典范"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "perfect standard of beauty, intellect etc.", "word": "理想"}, {"code": "cs", "lang": "Czech", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideál"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ideaal"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["feminine"], "word": "perfectie"}, {"code": "nl", "lang": "Dutch", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "streefdoel"}, {"code": "eo", "lang": "Esperanto", "sense": "perfect standard of beauty, intellect etc.", "word": "idealo"}, {"code": "eo", "lang": "Esperanto", "sense": "perfect standard of beauty, intellect etc.", "word": "perfekta"}, {"code": "fi", "lang": "Finnish", "sense": "perfect standard of beauty, intellect etc.", "word": "ihanne"}, {"code": "fi", "lang": "Finnish", "sense": "perfect standard of beauty, intellect etc.", "word": "ideaali"}, {"code": "fr", "lang": "French", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "idéal"}, {"code": "gl", "lang": "Galician", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "de", "lang": "German", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "Ideal"}, {"code": "el", "lang": "Greek", "roman": "ideódes", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ιδεώδες"}, {"code": "he", "lang": "Hebrew", "roman": "ide'ál", "sense": "perfect standard of beauty, intellect etc.", "word": "אידיאל \\ אִידֵאָל"}, {"code": "hi", "lang": "Hindi", "roman": "āiṛiyal", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "आइड़ियल"}, {"code": "hi", "lang": "Hindi", "roman": "ādarś", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "आदर्श"}, {"code": "hi", "lang": "Hindi", "roman": "śreṣṭh", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "श्रेष्ठ"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "eszmény"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "ideál"}, {"code": "hu", "lang": "Hungarian", "sense": "perfect standard of beauty, intellect etc.", "word": "mintakép"}, {"code": "io", "lang": "Ido", "sense": "perfect standard of beauty, intellect etc.", "word": "idealo"}, {"code": "id", "lang": "Indonesian", "sense": "perfect standard of beauty, intellect etc.", "word": "ideal"}, {"code": "ia", "lang": "Interlingua", "sense": "perfect standard of beauty, intellect etc.", "word": "ideal"}, {"code": "ga", "lang": "Irish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "idéal"}, {"code": "it", "lang": "Italian", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideale"}, {"alt": "りそう", "code": "ja", "lang": "Japanese", "roman": "risō", "sense": "perfect standard of beauty, intellect etc.", "word": "理想"}, {"code": "kk", "lang": "Kazakh", "roman": "mūrat", "sense": "perfect standard of beauty, intellect etc.", "word": "мұрат"}, {"alt": "理想", "code": "ko", "lang": "Korean", "roman": "isang", "sense": "perfect standard of beauty, intellect etc.", "word": "이상"}, {"alt": "理想", "code": "ko", "lang": "Korean", "roman": "risang", "sense": "perfect standard of beauty, intellect etc.", "tags": ["North-Korea"], "word": "리상"}, {"code": "fa", "lang": "Persian", "roman": "ârmân", "sense": "perfect standard of beauty, intellect etc.", "word": "آرمان"}, {"code": "fa", "lang": "Persian", "roman": "ide'âl", "sense": "perfect standard of beauty, intellect etc.", "word": "ایدئال"}, {"code": "pl", "lang": "Polish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideał"}, {"code": "pt", "lang": "Portuguese", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "pa", "lang": "Punjabi", "roman": "ādarś", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ਆਦਰਸ਼"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "идеа́л"}, {"code": "sk", "lang": "Slovak", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideál"}, {"code": "sl", "lang": "Slovene", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "es", "lang": "Spanish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ideal"}, {"code": "sv", "lang": "Swedish", "sense": "perfect standard of beauty, intellect etc.", "tags": ["neuter"], "word": "ideal"}, {"code": "te", "lang": "Telugu", "roman": "ādarśaṁ", "sense": "perfect standard of beauty, intellect etc.", "word": "ఆదర్శం"}, {"code": "uk", "lang": "Ukrainian", "roman": "ideál", "sense": "perfect standard of beauty, intellect etc.", "tags": ["masculine"], "word": "ідеа́л"}, {"code": "vi", "lang": "Vietnamese", "sense": "perfect standard of beauty, intellect etc.", "word": "lý tưởng"}, {"code": "cy", "lang": "Welsh", "sense": "perfect standard of beauty, intellect etc.", "tags": ["feminine", "masculine"], "word": "delfryd"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "subring closed under multiplication by containing ring", "word": "理想"}, {"code": "nl", "lang": "Dutch", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideaal"}, {"code": "fi", "lang": "Finnish", "sense": "subring closed under multiplication by containing ring", "word": "ideaali"}, {"code": "fr", "lang": "French", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "idéal"}, {"code": "de", "lang": "German", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "Ideal"}, {"code": "ja", "lang": "Japanese", "roman": "idearu", "sense": "subring closed under multiplication by containing ring", "word": "イデアル"}, {"code": "kk", "lang": "Kazakh", "roman": "ideal", "sense": "subring closed under multiplication by containing ring", "word": "идеал"}, {"code": "pl", "lang": "Polish", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "ideał"}, {"code": "ro", "lang": "Romanian", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideal"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "subring closed under multiplication by containing ring", "tags": ["masculine"], "word": "идеа́л"}, {"code": "sv", "lang": "Swedish", "sense": "subring closed under multiplication by containing ring", "tags": ["neuter"], "word": "ideal"}, {"code": "cmn", "lang": "Chinese Mandarin", "roman": "lǐxiǎng", "sense": "lower set closed under joins", "word": "理想"}, {"code": "ga", "lang": "Irish", "sense": "lower set closed under joins", "tags": ["masculine"], "word": "idéal"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "lower set closed under joins", "tags": ["masculine"], "word": "идеа́л"}, {"code": "de", "lang": "German", "sense": "unattainable state", "tags": ["masculine"], "word": "Idealzustand"}, {"code": "pl", "lang": "Polish", "sense": "unattainable state", "tags": ["masculine"], "word": "ideał"}, {"code": "ru", "lang": "Russian", "roman": "ideál", "sense": "unattainable state", "tags": ["masculine"], "word": "идеа́л"}], "wikipedia": ["Ideal", "Richard Dedekind"], "word": "ideal"}

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