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

A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
(algebra, ring theory) A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
(algebra, order theory, lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
(set theory) 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.
(algebra, 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.
(algebra) 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.

There will always be a wide interval between practical and ideal excellence.
The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors —
Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
the ideal theory or philosophy
ideal point
An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.
If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an ideal.
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 - Carl Schurz
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}.

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