ideals background

ideals

/aɪˈdɪəlz/ /aɪˈdi.əlz/

Illustration ideals - Image 1

Illustration ideals - Image 2

noun

Ideale, Leitbilder

A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.

Examples :

"Even though it's hard, Maria strives to live up to her ideals of honesty and kindness in her everyday actions. "

Obwohl es schwer ist, bemüht sich Maria, ihren Idealen von Ehrlichkeit und Freundlichkeit in ihren täglichen Handlungen gerecht zu werden.

value philosophy moral quality attitude being character

Illustration ideals - Image 1

noun

Ideal

A subring closed under multiplication by its containing ring.

Examples :

"In abstract algebra, ideals are special subsets within a ring that absorb multiplication from the larger ring. "

In der abstrakten Algebra sind Ideale spezielle Teilmengen innerhalb eines Rings, die die Multiplikation vom größeren Ring absorbieren.

Illustration ideals - Image 1

noun

Ideal

(lattice theory) A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).

Examples :

"The study of ideals in lattice theory helps mathematicians understand how sets of elements are organized and related based on specific rules of inclusion and combination. "

Das Studium von Idealen in der Verbandstheorie hilft Mathematikern zu verstehen, wie Mengen von Elementen auf der Grundlage spezifischer Regeln der Inklusion und Kombination organisiert und miteinander verbunden sind.

theory math logic system

Illustration ideals - Image 1

noun

Ideale

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.

Examples :

"The mathematician studied ideals of numbers, focusing on collections where adding or reducing elements always resulted in another member of the same collection. "

Der Mathematiker studierte Ideale von Zahlen und konzentrierte sich dabei auf Mengen, bei denen das Hinzufügen oder Reduzieren von Elementen immer zu einem anderen Mitglied derselben Menge führte.

Illustration ideals - Image 1

noun

Ideal

(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.

Examples :

"Within the Lie algebra that describes a rotating object's movements, the ideals represent specific ways the object can rotate that, when combined with any other possible movement of the object, still result in one of those specific rotations. "

Innerhalb der Lie-Algebra, die die Bewegungen eines rotierenden Objekts beschreibt, stellen die Ideale spezifische Arten dar, wie sich das Objekt drehen kann, die, wenn sie mit jeder anderen möglichen Bewegung des Objekts kombiniert werden, immer noch zu einer dieser spezifischen Rotationen führen.