# ideal

## English

### Etymology

From French idéal, from Late Latin ideālis ‎(existing in idea), from Latin idea ‎(idea); see idea.

### Pronunciation

• Rhymes: -iːəl
• IPA(key): /aɪˈdɪəl/, /aɪˈdiː.əl/
•  Audio (US) (file)

ideal ‎(comparative more ideal, superlative most ideal)

1. Optimal; being the best possibility.
2. Perfect, flawless, having no defects.
• Rambler
There will always be a wide interval between practical and ideal excellence.
3. Pertaining to ideas, or to a given idea.
4. Existing only in the mind; conceptual, imaginary.
• 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 —
• 1818, Mary Shelley, Frankenstein, or the Modern Prometheus,[1] Chapter 4,
Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
5. Teaching or relating to the doctrine of idealism.
the ideal theory or philosophy
6. (mathematics) Not actually present, but considered as present when limits at infinity are included.
ideal point
An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

#### Translations

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### Noun

ideal ‎(plural ideals)

1. 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
2. (mathematics, order theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).[2]
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.
3. (for example, algebra) A subring closed under multiplication by its containing ring.
Let ${\displaystyle \mathbb {Z} }$ be the ring of integers and let ${\displaystyle 2\mathbb {Z} }$ be its ideal of even integers. Then the quotient ring ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ is a Boolean ring.
The product of two ideals ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ is an ideal ${\displaystyle {\mathfrak {ab}}}$ which is a subset of the intersection of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$. This should help to understand why maximal ideals are prime ideals. Likewise, the union of ${\displaystyle {\mathfrak {a}}}$ and ${\displaystyle {\mathfrak {b}}}$ is a subset of ${\displaystyle {\mathfrak {a+b}}}$.