ideal

see also: Ideal

Pronunciation

Proper noun

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see also: Ideal

Pronunciation

- IPA: /aɪˈdɪəl/, /aɪˈdiː.əl/

**ideal**

- Optimal; being the best possibility.
- Perfect, flawless, having no defects.
**1751**April 13, Samuel Johnson,*The Rambler*, Number 112, reprinted in 1825,*The Works of Samuel Johnson, LL. D.*, Volume 1, Jones & Company, page 194 ↗,- There will always be a wide interval between practical and
**ideal**excellence; […] .

- There will always be a wide interval between practical and

- Pertaining to ideas, or to a given idea.
- 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 —

- The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by
**1818**, Mary Shelley,*Frankenstein, or the Modern Prometheus*,^{}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.

- Life and death appeared to me

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

- See also Thesaurus:flawless

**ideal** (*plural* ideals)

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

- (
*algebra, ring theory*) 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.*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}.

**2004**, K. R. Goodearl, R. B. Warfield, Jr.,*An Introduction to Noncommutative Noetherian Rings*, 2nd Edition, Cambridge University Press, page 47 ↗,- In trying to understand the
**ideal**theory of a commutative ring, one quickly sees that it is important to first understand the prime**ideals**.

- In trying to understand the
**2009**, John J. Watkins,*Topics in Commutative Ring Theory*, Princeton University Press, page 45 ↗,- If an
**ideal***I*of a ring contains the multiplicative identity 1, then we have seen that*I*must be the entire ring.

- If an
**2010**, W. D. Burgess, A. Lashgari, A. Mojiri,*Elements of Minimal Prime*, Sergio R. López-Permouth, Dinh Van Huynh (editors),**Ideals**in General Rings*Advances in Ring Theory*, Springer (Birkhäuser), page 69 ↗,- However, every
*R*has a minimal prime**ideal**consisting of left zero-divisors and one of right zero-divisors.

- However, every

- (
*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).**1992**, Unnamed translator, T. S. Fofanova,*General Theory of Lattices*, in*Ordered Sets and Lattices II*, American Mathematical Society, page 119 ↗,- An
**ideal***A*of*L*is called complete if it contains all least upper bounds of its subsets that exist in*L*. Bishop and Schreiner [80] studied conditions under which joins of**ideals**in the lattices of all**ideals**and of all complete**ideals**coincide.

- An
**2011**, George Grätzer,*Lattice Theory: Foundation*, Springer (Birkhäuser), page 125 ↗,- 1.35 Find a distributive lattice
*L*with no minimal and no maximal prime**ideals**.

- 1.35 Find a distributive lattice
**2015**, Vijay K. Garg,*Introduction to Lattice Theory with Computer Science Applications*, Wiley, page 186 ↗,**Definition 15.11**(**Width Ideal**) ''An**ideal**Q of a poset P = (X,≤) is a width**ideal**if maximal(Q) is a width antichain.

- (
*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.*Formally, an***ideal**I of a given set X is a nonempty subset of the powerset \mathcal{P}(X) such that: (1)\ \emptyset \in I, (2)\ A \in I \and B \subseteq A\implies B\in I and (3)\ A,B \in I\implies A\cup B \in I.

- (
*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.**1975**, Che-Young Lee (translator), Zhe-Xian Wan,*Lie Algebras*, Pergamon Press, page 13 ↗,- If 𝖌 is a Lie algebra, 𝖍 is an
**ideal**and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.

- If 𝖌 is a Lie algebra, 𝖍 is an
**2006**, W. McGovern,*The work of Anthony Joseph in classical representation theory*, Anthony Joseph, Joseph Bernstein, Vladimir Hinich, Anna Melnikov (editors),*Studies in Lie Theory: Dedicated to A. Joseph on His Sixtieth Birthday*, Springer (Birkhäuser), page 3 ↗,- What really put primitive
**ideals**in enveloping algebras of semisimple Lie algebras on the map was Duflo's fundamental theorem that any such**ideal**is the annihilator of a very special kind of simple module, namely a highest weight module.

- What really put primitive
**2013**, J.E. Humphreys,*Introduction to Lie Algebras and Representation Theory*, Springer, page 73 ↗,- Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L_1\oplus \dots \oplus L_t of simple
**ideals**(Theorem 5.2).

- Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L_1\oplus \dots \oplus L_t of simple

- (
*type of Lie subalgebra*) Lie ideal

- (
*order theory*) filter

- Russian: идеа́л

- German: Idealzustand
- Russian: идеа́л

**Ideal**Proper noun

- A city in Georgia, USA.
- An unincorporated community in Illinois.
- An unincorporated community in South Dakota.

This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.004