different between ideal vs side

English

Etymology

From French idéal, from Late Latin ide?lis (“existing in idea”), 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

• Rhymes: -i??l
• IPA^(key): /a??d??l/, /a??di?.?l/

Adjective

ideal (comparative more ideal, superlative most ideal)

1. Optimal; being the best possibility.
2. 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; […] .

3. Pertaining to ideas, or to a given idea.
4. Existing only in the mind; conceptual, imaginary.
   • 1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:

     The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —

   • 1818, Mary Shelley, Frankenstein, or the Modern Prometheus,^[1] Chapter 4,

     Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.

5. Teaching or relating to the doctrine of idealism.

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

Synonyms

• See also Thesaurus:flawless

Related terms

Derived terms

Translations

Noun

ideal (plural ideals)

2. 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

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

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

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

   • 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd Edition, Cambridge University Press, page 47,

     In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.

   • 2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45,

     If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.

   • 2010, W. D. Burgess, A. Lashgari, A. Mojiri, Elements of Minimal Prime Ideals in General Rings, Sergio R. López-Permouth, Dinh Van Huynh (editors), Advances in Ring Theory, Springer (Birkhäuser), page 69,

     However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.

4. (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.

   • 2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125,

     1.35 Find a distributive lattice L with no minimal and no maximal prime ideals.

   • 2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186,

     Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,?) is a width ideal if maximal(Q) is a width antichain.

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

6. (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 ????.
   • 1975, Che-Young Lee (translator), Zhe-Xian Wan, Lie Algebras, Pergamon Press, page 13,

     If ???? is a Lie algebra, ???? is an ideal and the Lie algebras ???? and ????/???? are solvable, then ???? is solvable.

   • 2006, W. McGovern, The work of Anthony Joseph in classical representation theory, 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,

     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.

   • 2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73,

     Next let ${\displaystyle L}$ be an arbitrary semisimple Lie algebra. Then ${\displaystyle L}$ can be written uniquely as a direct sum ${\displaystyle L_{1}\oplus \dots \oplus L_{t}}$ of simple ideals (Theorem 5.2).

Synonyms

• (type of Lie subalgebra): Lie ideal

Antonyms

• (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#Subalgebras, ideals and homomorphisms on Wikipedia.Wikipedia

Anagrams

• Delia, Elida, ailed, ladie

Asturian

Etymology

From Latin ide?lis.

Pronunciation

Adjective

ideal (epicene, plural ideales)

1. ideal

Noun

ideal m (plural ideales)

1. ideal

Catalan

Etymology

From Latin ide?lis.

Pronunciation

• (Balearic, Central, Valencian) IPA^(key): /i.de?al/

Adjective

ideal (masculine and feminine plural ideals)

1. ideal

Derived terms

• idealment

Noun

ideal m (plural ideals)

1. ideal

Galician

Etymology

From Latin ide?lis.

Pronunciation

Adjective

ideal m or f (plural ideais)

1. ideal

Derived terms

• idealmente

Noun

ideal m (plural ideais)

1. ideal

German

Etymology

Borrowed from Late Latin ide?lis (“existing in idea”), from Latin idea (“idea”). Doublet of ideell.

Pronunciation

• IPA^(key): /ide?a?l/
• Rhymes: -a?l

Adjective

ideal (comparative idealer, superlative am idealsten)

1. 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 ide?lis.

Pronunciation

• IPA^(key): [??d?eal]
• Hyphenation: idé?al

Adjective

ideal

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

Noun

ideal (plural ideal-ideal, first-person possessive idealku, second-person possessive idealmu, third-person possessive idealnya)

1. (mathematics) ideal: a subring closed under multiplication by its containing ring.

Alternative forms

• idiil

Affixed terms

Related terms

Further reading

• “ideal” in Kamus Besar Bahasa Indonesia (KBBI) Daring, Jakarta: Badan Pengembangan dan Pembinaan Bahasa, Kementerian Pendidikan dan Kebudayaan Republik Indonesia, 2016.

Luxembourgish

Pronunciation

• IPA^(key): /ide?a?l/, [id??a?l], /idi?a?l/

Adjective

ideal (masculine idealen, neuter ideaalt, comparative méi ideal, superlative am ideaalsten)

1. ideal

Declension

Norwegian Bokmål

Etymology

From French idéal, from Late Latin ide?lis (“existing in idea”), from Latin idea (“idea”)

Noun

ideal n (definite singular idealet, indefinite plural ideal or idealer, definite plural ideala or idealene)

1. ideal

References

• “ideal” in The Bokmål Dictionary.

Norwegian Nynorsk

Etymology

From French idéal, from Late Latin ide?lis (“existing in idea”), from Latin idea (“idea”)

Noun

ideal n (definite singular idealet, indefinite plural ideal, definite plural ideala)

1. ideal

References

• “ideal” in The Nynorsk Dictionary.

Piedmontese

Pronunciation

• IPA^(key): /ide?al/

Adjective

ideal

1. ideal

Noun

ideal m

1. ideal

Portuguese

Etymology

From Latin ide?lis.

Pronunciation

• (Portugal) IPA^(key): /i?ðja?/
• (Brazil) IPA^(key): /i.de?aw/
• Hyphenation: i?de?al

Adjective

ideal m or f (plural ideais, comparable)

1. ideal
2. notional

Derived terms

• idealmente

Noun

ideal m (plural ideais)

1. ideal
2. fantasy

Romanian

Etymology

From French idéal, from Latin idealis.

Adjective

ideal m or n (feminine singular ideal?, masculine plural ideali, feminine and neuter plural ideale)

1. ideal

Declension

Related terms

• idealism
• idealist
• idealiza

Serbo-Croatian

Pronunciation

• IPA^(key): /id?a?l/
• Hyphenation: i?de?al

Noun

idè?l m (Cyrillic spelling ??????)

1. ideal

Declension

Spanish

Etymology

From Latin ide?lis.

Pronunciation

• IPA^(key): /ide?al/, [i.ð?e?al]

Adjective

ideal (plural ideales)

1. ideal

Derived terms

Noun

ideal m (plural ideales)

1. ideal

Swedish

Pronunciation

Noun

ideal n

1. ideal; perfect standard
2. (mathematics) ideal; special subsets of a ring

Declension

Anagrams

• ilade

Turkish

Etymology

Borrowed from French idéal.

Pronunciation

• IPA^(key): /ide?a?/
• Hyphenation: i?de?al

Adjective

ideal (comparative daha ideal, superlative en ideal)

1. ideal

Noun

ideal (definite accusative ideali, plural idealler)

1. ideal

Declension

Synonyms

• ülkü

English

Pronunciation

• enPR: s?d, IPA^(key): /sa?d/
• Hyphenation: side
• Rhymes: -a?d
• Homophone: sighed

Etymology 1

From Middle English side, from Old English s?de (“side, flank”), from Proto-Germanic *s?d? (“side, flank, edge, shore”), from Proto-Indo-European *s?y- (“to send, throw, drop, sow, deposit”). Cognate with Saterland Frisian Siede (“side”), West Frisian side (“side”), Dutch zijde, zij (“side”), German Low German Sied (“side”), German Seite (“side”), Danish and Norwegian side (“side”), Swedish sida (“side”).

Noun

side (countable and uncountable, plural sides)

1. A bounding straight edge of a two-dimensional shape.

2. A flat surface of a three-dimensional object; a face.

3. One half (left or right, top or bottom, front or back, etc.) of something or someone.
4. A region in a specified position with respect to something.
5. The portion of the human torso usually covered by the arms when they are not raised; the areas on the left and right between the belly or chest and the back.
   • 2006, American Academy of Orthopaedic Surgeons, Emergency Care and Transportation of the Sick and Injured (Jones & Bartlett Learning, ?ISBN, p. 234:

     Roll the patient onto the left side so that head, shoulders, and torso move at the same time without twisting.

6. One surface of a sheet of paper (used instead of "page", which can mean one or both surfaces.)

7. One possible aspect of a concept, person or thing.

8. One set of competitors in a game.

9. (Britain, Australia, Ireland) A sports team.
   • 2011, Nick Cain, Greg Growden, Rugby Union For Dummies, UK Edition, 3rd Edition, p.220:

     Initially, the English, Welsh, Scots and Irish unions refused to send national sides, preferring instead to send touring sides like the Barbarians, the Penguins, the Co-Optimists, the Wolfhounds, Crawshays Welsh, and the Public School Wanderers.

10. A group of morris dancers who perform together.
11. A group having a particular allegiance in a conflict or competition.
    • 2019, VOA Learning English (public domain)

      “Creating artificial rain over the Yellow Sea would help the Chinese side too,” the spokesman said Kim told the meeting.

    • 1824, Walter Savage Landor, Imaginary Conversations, Lord Chesterfield and Lord Chatham

      We have not always been of the […] same side in politics.

    • sets the passions on the side of truth

12. (music) A recorded piece of music; a record, especially in jazz.
    • 1995, James Lincoln Collier, Jazz: The American Theme Song, p. 41

      But Bechet chafed under even the loose discipline of the Ellington group, and left. Through these years he wandered, making only a few sides, at the moment when jazz records were beginning to flood onto the market.

13. (sports, billiards, snooker, pool) Sidespin; english

14. (Britain, Australia, Ireland, dated) A television channel, usually as opposed to the one currently being watched (from when there were only two channels).

15. (US, colloquial) A dish that accompanies the main course; a side dish.

16. A line of descent traced through one parent as distinguished from that traced through another.
17. (baseball) The batters faced in an inning by a particular pitcher

    Clayton Kershaw struck out the side in the 6th inning.

18. (slang, dated, uncountable) An unjustified air of self-importance.
19. (drama) A written monologue or part of a scene to be read by an actor at an audition.
    • 2010, Viola Spolin, ?Carol Sills, Theater Games for Rehearsal: A Director's Handbook (page 12)

      Some directors use full scripts (book); others use “sides,” which consist of one or two words of the cue and the subsequent full speech of the individual actor.
20. (LGBT, slang) A man who prefers not to engage in anal sex during homosexual intercourse.

    My boyfriend and I are both sides, so we prefer to do oral on each other.

Synonyms

• (bounding straight edge of an object): edge
• (flat surface of an object): face
• (left or right half): half
• (surface of a sheet of paper): page
• (region in a specified position with respect to something):
• (one possible aspect of a concept):
• (set of opponents in a game): team
• (group having a particular allegiance in a war):
• (television channel): channel, station (US)

Hyponyms

Derived terms

Related terms

Translations

Adjective

side (comparative more side, superlative most side)

1. Being on the left or right, or toward the left or right; lateral.

   • One mighty squadron with a side wind sped.

2. Indirect; oblique; incidental.

   a side issue; a side view or remark

   • 1594, Richard Hooker, Of the Lawes of Ecclesiastical Politie

     The law hath no side respect to their persons.

Verb

side (third-person singular simple present sides, present participle siding, simple past and past participle sided)

1. (intransitive) To ally oneself, be in an alliance, usually with "with" or rarely "in with"

   Which will you side with, good or evil?

   • 1597, Francis Bacon, Essays – "Of Great Place":

     All rising to great place is by a winding star; and if there be factions, it is good to side a man's self, whilst he is in the rising, and to balance himself when he is placed.

   • 1958, Archer Fullingim, The Kountze [Texas] News, August 28, 1958:

     How does it feel... to... side in with those who voted against you in 1947?

2. To lean on one side.

   (Can we find and add a quotation of Francis Bacon to this entry?)

3. (transitive, obsolete) To be or stand at the side of; to be on the side toward.
4. (transitive, obsolete) To suit; to pair; to match.

   (Can we find and add a quotation of Clarendon to this entry?)

5. (transitive, shipbuilding) To work (a timber or rib) to a certain thickness by trimming the sides.
6. (transitive) To furnish with a siding.

   to side a house

7. (transitive, cooking) To provide with, as a side or accompaniment.
   • 1995, Orange Coast Magazine (volume 11, number 8, page 166)

     Entrees are sided with a generous portion of vegetables, and some include little surprises […]

Synonyms

• (ally oneself):
• take side

Derived terms

• side with
• siding

Translations

See also

• ally
• alliance
• join in

Etymology 2

From Middle English side, syde, syd, from Old English s?d (“wide, broad, spacious, ample, extensive, vast, far-reaching”), from Proto-Germanic *s?daz (“drooping, hanging, low, excessive, extra”), from Proto-Indo-European *s?y- (“to send, throw, drop, sow, deposit”). Cognate with Low German sied (“low”), Swedish sid (“long, hanging down”), Icelandic síður (“low hanging, long”).

Adjective

side (comparative more side, superlative most side)

1. (Britain archaic, dialectal, Northern England, Scotland) Wide; large; long, pendulous, hanging low, trailing; far-reaching.
   • c. 1556, Thomas Cranmer, A Confutation of Unwritten Verities, “That the general counsels withoute the worde of god are not sufficiente to make articles of fayth,”^[2]

     But when he perceaved that the sayd Pryest could not pourge himself of the foresayd crime he prively payed him his quarters wages before hande and suffered hym to departe without farther tryall of the sayd cryme: and now he jetteth in london wyth side gown and sarcenet typet as good a virgin priest as the best.

   • 1575, Robert Laneham, Robert Laneham’s Letter: Describing a Part of the Entertainment unto Queen Elizabeth at the Castle of Kenelworth in 1575, edited by F. J. Furnivall, London: Chatto & Windus, 1907, “The auncient Minstrell described,” p. 38,^[3]

     Hiz gooun had syde sleeuez dooun to midlegge, slit from the shooulder too the hand, & lined with white cotten.

   • 1595, George Peele, The Old Wives’ Tale, The Malone Society Reprints, 1908, lines 47-50,^[4]

     What doe we make dost thou aske? why we make faces for feare: such as if thy mortall eyes could behold, would make thee water the long seames of thy side slops […]

   • c. 1598, William Shakespeare, Much Ado about Nothing, Act III, Scene 4,^[5]

     By my troth, ’s but a night-gown in respect of yours: cloth o’ gold, and cuts, and laced with silver, set with pearls, down sleeves, side sleeves, and skirts, round underborne with a bluish tinsel […]

2. (Scotland) Far; distant.

Derived terms

• sidth

Adverb

side (comparative more side, superlative most side)

1. (Britain dialectal) Widely; wide; far.

Verb

side (third-person singular simple present sides, present participle siding, simple past and past participle sided)

1. To clear, tidy or sort.

Anagrams

• Desi, Dies, EIDs, Eids, IDEs, IEDs, Ides, SEID, deis, desi, dies, eids, ides, sied

Danish

Etymology

From Old Norse síða.

Pronunciation

• IPA^(key): /?si?d?/, [?si?ð??]
• Rhymes: -i?d?

Noun

side c (singular definite siden, plural indefinite sider)

1. page

Declension

Further reading

• “side” in Den Danske Ordbog
• “side” in Ordbog over det danske Sprog

Estonian

Etymology

From Proto-Finnic *sidek. Equivalent to siduma +? -e.

Noun

side (genitive sideme, partitive sidet)

1. bond, binding
2. bandage
3. relationship, tie

Inflection

Compounds

• kaelaside

Noun

side (genitive side, partitive sidet)

1. communication (especially one achieved through technology)
2. signal (especially in radio)

3. communications (as a field)
4. (colloquial) post office

Inflection

Compounds

• otseside

Finnish

(index si)

Etymology

From Proto-Finnic *sidek. Equivalent to sitoa +? -e.

Pronunciation

• IPA^(key): /?side?/, [?s?ide?(?)]
• Rhymes: -ide
• Syllabification: si?de

Noun

side

1. bandage
2. bond
3. sanitary towel
4. (anatomy) ligament

Declension

Synonyms

• (sanitary towel): terveysside
• (ligament): ligamentti

Derived terms

Related terms

• sidos

Anagrams

• desi, desi-

Latin

Verb

s?de

1. second-person singular present active imperative of s?d?

Manx

Etymology

From Old Irish saiget, from Latin sagitta.

Noun

side f (genitive singular sidey, plural sideyn)

1. arrow, bolt, shaft

Related terms

• fleit
• sideyr (“archer”)

Mutation

References

• Gregory Toner, Maire Ní Mhaonaigh, Sharon Arbuthnot, Dagmar Wodtko, Maire-Luise Theuerkauf, editors (2019) , “saiget”, in eDIL: Electronic Dictionary of the Irish Language

Middle Irish

Etymology

From Old Irish síd, from Proto-Celtic *sedos, *s?dos (“mound (inhabited by fairies)”), from Proto-Indo-European *s?ds, *sed- (“seat”).

Noun

side m

1. fairy hill or mound

Derived terms

Descendants

• Irish: sí

Mutation

References

• Gregory Toner, Maire Ní Mhaonaigh, Sharon Arbuthnot, Dagmar Wodtko, Maire-Luise Theuerkauf, editors (2019) , “1 síd, síth”, in eDIL: Electronic Dictionary of the Irish Language

Norwegian Bokmål

Etymology

From Old Norse síða.

Noun

side f or m (definite singular sida or siden, indefinite plural sider, definite plural sidene)

1. a page (e.g. in a book)
2. side

3. (of a case) aspect
4. (on animal) flank

Derived terms

Norwegian Nynorsk

Pronunciation

• IPA^(key): /²si?(d)?/ (examples of pronunciation)

Etymology 1

From Old Norse síða. Akin to English side.

Noun

side f (definite singular sida, indefinite plural sider, definite plural sidene)

1. a page (e.g. in a book)

2. a side (various, though not all senses)

Derived terms

Etymology 2

Adjective

side

1. definite singular of sid
2. plural of sid

References

• “side” in The Nynorsk Dictionary.

Anagrams

• deis, desi-, dise, seid

Old English

Pronunciation

• IPA^(key): /?si?.de/

Etymology 1

From the adjective s?d.

Adverb

s?de

1. widely

Etymology 2

From Proto-Germanic *s?d?, whence also Old High German s?ta

Noun

s?de f

1. side

Declension

Etymology 3

Borrowed from Late Latin s?ta, whence also Old High German s?da (“silk”).

Noun

s?de f (nominative plural s?dan)

1. silk

Synonyms

• seolc

Old Irish

Pronunciation

• IPA^(key): /?s?ið?e/

Pronoun

side

1. inflection of suide:
   1. nominative/accusative singular masculine unstressed
   2. genitive singular feminine unstressed

Mutation

West Frisian

Etymology

From Old Frisian s?de, from Proto-Germanic *s?d?.

Pronunciation

• IPA^(key): /?sid?/

Noun

side c (plural siden, diminutive sydsje)

1. side
2. page

Derived terms

• webside

Further reading

• “side (I)”, in Wurdboek fan de Fryske taal (in Dutch), 2011