ideal

Etymology

From French idéal, from Late Latin ideālis (“existing in idea”), from Latin idea (“idea”); see idea. In mathematics, the noun ring theory sense was first introduced by German mathematician Richard Dedekind in his 1871 edition of a text on number theory. The concept was quickly expanded to ring theory and later generalised to order theory. The set theory and Lie theory senses can be regarded as applications of the order theory sense.

adj

1. Optimal; being the best possibility.
2. 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; […] .
3. Pertaining to ideas, or to a given idea.
4. Existing only in the mind; conceptual, imaginary.
   The idea of ghosts is ridiculous in the extreme; and if you continue to be swayed by ideal terrors 1796, Matthew Lewis, The Monk, Folio Society, published 1985, page 256
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.

noun

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
   With great humility, I call upon all Americans to help me keep our nation united in defense of those ideals which have been so eloquently proclaimed by Franklin Roosevelt. I want in turn to assure my fellow Americans and all of those who love peace and liberty throughout the world that I will support and defend those ideals with all my strength and all my heart. 16 April 1945, Harry S. Truman, 9:21 from the start, in MP72-20 President Roosevelt’s Funeral and Procession; Truman – New President of U.S., Harry S. Truman Presidential Library and Museum, National Archives Identifier: 595162
2. (algebra, ring theory) A subring closed under multiplication by its containing ring.
   Let ℤ be the ring of integers and let 2ℤ be its ideal of even integers. Then the quotient ring ℤ/2ℤ is a Boolean ring.
   The product of two ideals a and b is an ideal ab which is a subset of the intersection of a and b. This should help to understand why maximal ideals are prime ideals. Likewise, the union of a and b is a subset of a+b.
   In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, 2nd edition, Cambridge University Press, page 47
   If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring. 2009, John J. Watkins, Topics in Commutative Ring Theory, Princeton University Press, page 45
   However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors. 2010, W. D. Burgess, A. Lashgari, A. Mojiri, “Elements of Minimal Prime Ideals in General Rings”, in Sergio R. López-Permouth, Dinh Van Huynh, editors, Advances in Ring Theory, Springer (Birkhäuser), page 69
3. (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.
   1.35 Find a distributive lattice L with no minimal and no maximal prime ideals. 2011, George Grätzer, Lattice Theory: Foundation, Springer (Birkhäuser), page 125
   Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain. 2015, Vijay K. Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, page 186
4. (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 𝒫(X) such that: (1)∅∈I, (2)A∈I and B⊆A⟹B∈I and (3)A,B∈I⟹A∪B∈I.
5. (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.
   If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable. 1975, Zhe-Xian Wan, translated by Che-Young Lee, Lie Algebras, Pergamon Press, page 13
   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. 2006, W. McGovern, “The work of Anthony Joseph in classical representation theory”, in 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
   Next let L be an arbitrary semisimple Lie algebra. Then L can be written uniquely as a direct sum L_1⊕…⊕L_t of simple ideals (Theorem 5.2). 2013, J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, page 73
6. (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.
   The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.