ideal

Meaning

Adj

1. Pertaining to ideas, or to a given idea.
2. Existing only in the mind; conceptual, imaginary.
3. Optimal; being the best possibility.
4. Perfect, flawless, having no defects.
5. Teaching or relating to the doctrine of idealism.
6. Not actually present, but considered as present when limits at infinity are included.

Noun

1. A thing which exists in the mind but not in reality; in ontological terms, a thing which has essence but not existence.
2. A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
3. A subring closed under multiplication by its containing ring.
4. A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
5. 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.
6. 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.
7. 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.

Pronounced as (IPA)

/aɪˈdiːl/

Etymology

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, 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.