Ideal

From French idÃ©al, from Late Latin ideÄlis ("existing in idea"), from Latin idea ("idea"); see idea.

Full definition of ideal

RamblerThere will always be a wide interval between practical and ideal excellence.

1796, Matthew Lewis, The Monk, Folio Society 1985, p. 256:The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors â€”
1818, Mary Shelley, ,^{http://en.wikisource.org/wiki/Frankenstein} ,Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.

Teaching or relating to the doctrine of idealism.the ideal theory or philosophy
(mathematics) Not actually present, but considered as present when limits at infinity are included.ideal pointAn ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

(plural ideals)

(a perfect standard of beauty, intellect etc.)A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.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
(mathematics, order theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).^{http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem#Prime_ideal_theorems}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.
(for example, algebra) A subring closed under multiplication by its containing ring.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.

\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}