ideal
suomi-englanti sanakirjaideal englannista suomeksi
ideaali, ihanne
ihanne-
ideaali-, ihanteellinen
esikuva
ideaalinen
Substantiivi
ideal englanniksi
Pertaining to ideas, or to a given idea.
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
(RQ:Mary Shelley Frankenstein)
(RQ:Dickens Pickwick Papers)
Optimal; being the best possibility.
1751 April 13, (w), ''(w)'', Number 112, reprinted in 1825, ''The Works of Samuel Johnson, LL. D.'', Volume 1, Jones & Company, page 194,
- There will always be a wide interval between practical and ideal excellence;(..).
Teaching or relating to the doctrine of idealism.
(ux)
Not actually present, but considered as present when limits at infinity are included.
A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.
(senseid)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'' - Schurz|Carl Schurz
{{quote-av
A subring closed under multiplication by its containing ring.
''Let \mathbb{Z} be the ring of integers and let 2\mathbb{Z} be its ideal of even integers. Then the quotient ring \mathbb{Z} / 2\mathbb{Z} is a Boolean ring.''
''The product of two ideals \mathfrak{a} and \mathfrak{b} is an ideal \mathfrak{a b} which is a subset of the intersection of \mathfrak{a} and \mathfrak{b}. This should help to understand why maximal ideals are prime ideals. Likewise, the union of \mathfrak{a} and \mathfrak{b} is a subset of \mathfrak{a + b}.''
{{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-VQKHVHvDWIQ6AEIngEwFQv=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=47|publisher=Cambridge University Press
{{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-VQKHVHvDWIQ6AEIfDAPv=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=45|publisher=Princeton University Press
{{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-VQKHVHvDWIQ6AEI7QEwIgv=onepage&q=%22ideal%22%7C%22ideals%22%20ring&f=false|page=69
A non-set|empty set (of a ordered set) which is closed under binary suprema (a.k.a. joins).(pedia)
1992, Unnamed translator, T. S. Fofanova, ''General Theory of Lattices'', in ''Ordered Sets and Lattices II'', (w), page 119,
- An ideal ''A'' of ''L'' is called complete if it contains all least upper bounds of its subsets that exist in ''L''. Bishop and Schreiner 80 studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.
{{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=0ahUKEwiHlaLcrtXdAhXNAYgKHYOxCe0Q6AEILTABv=onepage&q=%22ideal%22%7C%22ideals%22%20lattice&f=false|page=125
{{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=0ahUKEwiHlaLcrtXdAhXNAYgKHYOxCe0Q6AEIPjAEv=onepage&q=%22ideal%22%7C%22ideals%22%20lattice&f=false|page=186|publisher=Wiley
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 I of a given set X is a nonempty subset of the powerset \mathcal{P}(X) such that: (1)\ \emptyset \in I, (2)\ A \in I \and B \subseteq A\implies B\in I and (3)\ A,B \in I\implies A\cup B \in I.''
A subalgebra (subspace that is closed under the bracket) 𝖍 of a given 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-Q5dPdAhXQMt4KHQKvDPYQ6AEIRDAFv=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=13
{{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-Q5dPdAhXQMt4KHQKvDPYQ6AEISjAGv=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=3
{{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-Q5dPdAhXQMt4KHQKvDPYQ6AEIMzACv=onepage&q=%22ideal%22%7C%22ideals%22%20lie&f=false|page=73|publisher=Springer
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.(cite-web)|work=Chapter 1 : Lattice Theory|url=http://boole.stanford.edu/cs353/handouts/book1.pdf|publisher=boole.stanford.edu|section=§1.3.5
''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.''
(l)
ideal (gloss)
(l):
optimal; being the best possibility.
pertaining to ideas, or to a given idea.
(l): a subring closed under multiplication by its containing ring.
(l) (perfect)
(syn)
ideal; perfect standard
(l) (gl)