ideal
Noun pluralideals
Pluralideals
- 1Being the best possible; most suitable for a particular purpose.
The cabin's location is ideal for anyone who wants peace and quiet.
Spring offers ideal conditions for planting most vegetables.
Similarperfect, optimal, best, suitable, excellent
Oppositeunsuitable, poor, inappropriate
- 2
- 3A perfect standard of beauty, intellect, or excellence that one aims to reach.
The painting embodies a classical ideal of human beauty.
Their political ideals shaped every decision the group made.
Similarstandard, model, principle, value, paragon
Note
The plural is where this sense does its public work: ideals as the principles a person or a nation lives by. Speaking in 1945, Harry S. Truman asked Americans to stay united in defense of the ideals Franklin Roosevelt had proclaimed. Carl Schurz compared ideals to stars that guide a sailor even though no hand can touch them. People hold, proclaim and defend ideals; they rarely claim to reach them.
- 4Existing only in the mind; conceptual or imaginary rather than real.
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.
Similartheoretical, hypothetical, conceptual, imaginary, abstract
Oppositereal, actual, concrete, material
Note
Writers of the late 1700s and 1800s used it where a modern reader expects 'unreal'. Matthew Lewis's ideal terrors (1796) are fears of ghosts, and Mary Shelley's ideal bounds in Frankenstein are limits that exist only in thought. Dickens builds a joke on it: the goblins' kicking was certainly not ideal, because his aching shoulders proved it real. Older prose that sets the word against the real means this, not the 'best possible' meaning.
- 5Relating to ideas, or to a particular idea.
The critic examined the ideal content behind the poem's vivid images.
His philosophy treats ideal relations as distinct from material ones.
Similarconceptual
Oppositematerial
- 6Teaching or relating to the philosophical doctrine of idealism.
The ideal tradition holds that reality is fundamentally mental.
Critics dismissed his ideal reading of history as detached from material facts.
Oppositematerialist
- 7A subset of a ring closed under addition and under both left and right multiplication by elements of the ring; a two-sided ideal.
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.
Note
Ring theory is where the mathematical noun has its oldest roots: Richard Dedekind introduced it in his 1871 edition of a text on number theory. An ideal that contains 1 must be the whole ring, so the interesting ideals leave it out. Study of a ring usually starts with its prime ideals. Watch the attributive use: ideal theory means the theory of ideals, not a perfect theory.
- 8In mathematics, not actually present but treated as present once limits at infinity are included.
Parallel lines are said to intersect at an ideal point in projective geometry.
Adding ideal elements lets every pair of lines meet exactly once.
Note
A geometer adds ideal elements to supply what is missing. An ideal point lies at infinity, outside the plane proper, but it is counted as if it were there. In the hyperbolic disk, an ideal triangle has three geodesic sides that meet exactly on the boundary circle. Here the word always comes before a noun, as in ideal point, ideal triangle or ideal element, and it means 'adjoined at the limit', never 'perfect'.
- 9In order theory, a non-empty lower set of a partially ordered set that is closed under binary joins (suprema).
In a lattice, an ideal is downward closed and stable under finite joins.
Principal ideals are generated by taking everything below a fixed element.
Note
The order-theoretic counterpart of the ring-theory ideal is a set closed downward: anything below a member belongs to it, and the join of any two members stays inside. It usually comes with a qualifier, as in prime ideals, complete ideals, or the width ideals of lattice theory for computer science. George Grätzer's Lattice Theory: Foundation (2011) sets the task of finding a distributive lattice with no minimal and no maximal prime ideals.
- 10In set theory, a collection of sets regarded as small or negligible, closed under taking subsets of any member and unions of any 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.
Note
Set theory uses ideals to say which sets count as small, such as the finite ones or those of measure zero. Smallness passes downward, since any subset of a negligible set is negligible, and two small sets together remain small. The empty set always belongs. The mirror image is a filter, which gathers the large sets. Requiring closure under countable unions as well gives a σ-ideal.
- 11In Lie theory, a Lie subalgebra 𝖍 of a Lie algebra 𝖌 for which the bracket [𝖌, 𝖍] lies within 𝖍.
A simple Lie algebra has no proper nonzero ideal.
The center of a Lie algebra is always an ideal.
Note
Lie algebras use ideals the way rings do. An ideal 𝖍 is exactly what allows the quotient 𝖌/𝖍 to be formed, and ideals are the kernels of homomorphisms. Much of the structure theory runs through them. J.E. Humphreys's Introduction to Lie Algebras and Representation Theory (2013) writes every semisimple Lie algebra uniquely as a direct sum of simple ideals, and Duflo's theorem concerns primitive ideals in enveloping algebras.
- 12In algebra, a subsemigroup in which combining any element from outside it with one of its members yields a result lying outside the subsemigroup.
The proof verified that the chosen subsemigroup is indeed an ideal.
Researchers classified each ideal arising in the transformation semigroup.
Note
A semigroup has only one operation, so 'adding' here means combining by that operation, whatever it is. Take the natural numbers under multiplication. The powers of three, {1, 3, 9, 27, 81, …}, behave in this way: multiply one of them by a number that is not itself a power of three, and the product falls outside the set.
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Practice
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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.