Ideal
Last Updated: January 2, 2026 at 12:30 PM
Definition of Ideal
Adjective
Noun
- 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"
- A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
"Let Z {\displaystyle \mathbb {Z} } be the ring of integers and let 2 Z {\displaystyle 2\mathbb {Z} } be its ideal of even integers. Then the quotient ring Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } is a Boolean ring."
- A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
- 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 {\displaystyle I} of a given set X {\displaystyle X} is a nonempty subset of the powerset P ( X ) {\displaystyle {\mathcal {P}}(X)} such that: ( 1 ) ∅ ∈ I {\displaystyle (1)\ \emptyset \in I} , ( 2 ) A ∈ I ∧ B ⊆ A ⟹ B ∈ I {\displaystyle (2)\ A\in I\land B\subseteq A\implies B\in I} and ( 3 ) A , B ∈ I ⟹ A ∪ B ∈ I {\displaystyle (3)\ A,B\in I\implies A\cup B\in I} ."
- 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.
Anagrams of Ideal
Words You Can Make
28 words can be made from the letters in "ideal":