Meaning

• A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.

• A subring closed under multiplication by its containing ring.

• (lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).

• 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.

• (Lie theory) 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.

Meaning

• Optimal; being the best possibility.

• Perfect, flawless, having no defects.

• Pertaining to ideas, or to a given idea.

• Existing only in the mind; conceptual, imaginary.

• Teaching or relating to the doctrine of idealism.

  "the ideal theory or philosophy"

• Not actually present, but considered as present when limits at infinity are included.

  "An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle."