ideal UK /aɪˈdɪəl/US /aɪˈdi(ə)l/adjective 6noun 6forms 3 adjective6 •as good as it could possibly be for what is needed •Perfect, flawless, having no defects.There will always be a wide interval between practical and ideal excellence; […] . — 1751 •Pertaining to ideas, or to a given idea. •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 •Teaching or relating to the doctrine of idealism. •Not actually present, but considered as present when limits at infinity are included.An ideal triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle. noun6 •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 •A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.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 •A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).1.35 Find a distributive lattice L with no minimal and no maximal prime ideals. — 2011 •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. •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 •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. formsideals · more ideal · most ideal
