# ideal

Pronunciation: say "y-DEE-uhl" US /aɪˈdi(ə)l/, UK /aɪˈdɪəl/

## Adjective

1. Being the best possible; most suitable for a particular purpose.
   - Simple: The best you could possibly have or pick for a purpose.
   - _The cabin's location is ideal for anyone who wants peace and quiet._
   - _Spring offers ideal conditions for planting most vegetables._
2. Perfect and flawless; without any faults or defects.
   - Simple: Perfect, with nothing wrong or missing.
   - _Critics agreed the performance was close to ideal._
   - _He chased an ideal version of himself that no one could match._
4. Existing only in the mind; conceptual or imaginary rather than real.
   - Simple: Found only in the mind, not in the real world.
   - _The theory describes an ideal gas that has no exact real-world counterpart._
   - _He drew a sharp line between ideal forms and the objects we actually see._
5. Relating to ideas, or to a particular idea.
   - Simple: Having to do with ideas, or with one certain idea.
   - _The critic examined the ideal content behind the poem's vivid images._
   - _His philosophy treats ideal relations as distinct from material ones._
6. Teaching or relating to the philosophical doctrine of idealism.
   - Simple: Belonging to or teaching the idea-based view called idealism.
   - _The ideal tradition holds that reality is fundamentally mental._
   - _Critics dismissed his ideal reading of history as detached from material facts._
8. In mathematics, not actually present but treated as present once limits at infinity are included.
   - Simple: In math, not really there, but treated as present once points at infinity are added.
   - _Parallel lines are said to intersect at an ideal point in projective geometry._
   - _Adding ideal elements lets every pair of lines meet exactly once._

## Noun

3. A perfect standard of beauty, intellect, or excellence that one aims to reach.
   - Simple: A perfect example or standard that you try to reach.
   - _The painting embodies a classical ideal of human beauty._
   - _Their political ideals shaped every decision the group made._
7. A subset of a ring closed under addition and under both left and right multiplication by elements of the ring; a two-sided ideal.
   - Simple: In math, a part of a ring that stays inside itself when you multiply its members by any ring element.
   - _The kernel of a ring homomorphism is always an ideal of the domain ring._
   - _Quotient rings are built by collapsing an ideal down to zero._
9. In order theory, a non-empty lower set of a partially ordered set that is closed under binary joins (suprema).
   - Simple: In math, a set that keeps everything below its members and also holds the combination of any two of them.
   - _In a lattice, an ideal is downward closed and stable under finite joins._
   - _Principal ideals are generated by taking everything below a fixed element._
10. In set theory, a collection of sets regarded as small or negligible, closed under taking subsets of any member and unions of any two members.
   - Simple: In math, a group of sets treated as small, closed under taking parts and joining two members.
   - _The finite subsets of an infinite set form an ideal on that set._
   - _Measure-zero sets constitute an ideal that absorbs every smaller null set._
11. In Lie theory, a Lie subalgebra 𝖍 of a Lie algebra 𝖌 for which the bracket [𝖌, 𝖍] lies within 𝖍.
   - Simple: In advanced math, a part of a Lie algebra that stays inside itself when bracketed with any element.
   - _A simple Lie algebra has no proper nonzero ideal._
   - _The center of a Lie algebra is always an ideal._
12. In algebra, a subsemigroup in which combining any element from outside it with one of its members yields a result lying outside the subsemigroup.
   - Simple: In math, a part of a semigroup where adding any outside element to a member gives a result that stays outside.
   - _The proof verified that the chosen subsemigroup is indeed an ideal._
   - _Researchers classified each ideal arising in the transformation semigroup._

## Origin

From French idéal, from Late Latin ideālis ("existing in idea"), analysable as idea + -al, from Latin idea ("idea"). The mathematical ring-theory sense was introduced by the German mathematician Richard Dedekind in his 1871 edition of a text on number theory, and the concept was later expanded to ring theory and generalised to order theory.

Source: DictionaryAI, https://dictionaryai.com/define/ideal (written from Wiktionary data, CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0/).