ideals

Noun

ideal (plural ideals)

1. A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
   • Varvara Andreevna, when I was very young, I set before myself the ideal of the woman I loved and should be happy to call my wife.1901 [1878], chapter 5, in Constance Garnett, transl., Anna Karenina, translation of Анна Каренина (Anna Karenina) by Leo Tolstoy, part 6
2. (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.
   • Let #92;mathbb#123;Z#125; be the ring of integers and let 2#92;mathbb#123;Z#125; be its ideal of even integers. Then the quotient ring #92;mathbb#123;Z#125;#47;2#92;mathbb#123;Z#125; is a Boolean ring.
3. (lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
   • 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.1992, T. S. Fofanova, “General Theory of Lattices”, in Unnamed, transl., Ordered Sets and Lattices II, American Mathematical Society, page 119
4. 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 #92;mathcal#123;P#125;(X) such that: (1)#92;#92;emptyset#92;inI, (2)#92;A#92;inI#92;andB#92;subseteqA#92;impliesB#92;inI and (3)#92;A,B#92;inI#92;impliesA#92;cupB#92;inI.
5. (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

   Synonyms:

   Lie ideal

6. 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.

See all 15 meanings of ideal