ideals background

ideals

/aɪˈdɪəlz/ /aɪˈdi.əlz/

Illustration ideals - Image 1

Illustration ideals - Image 2

noun

理想, 理念

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. "

難しいことではあるが、マリアは毎日の行動において、誠実さと優しさという彼女の理想を追求している。

value philosophy moral quality attitude being character

Illustration ideals - Image 1

noun

イデアル

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. "

抽象代数学において、イデアルは、より大きな環からの乗算を吸収する環内の特別な部分集合です。

Illustration ideals - Image 1

noun

イデアル

(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. "

束論におけるイデアルの研究は、数学者が要素の集合が包含と組み合わせの特定の規則に基づいてどのように組織化され、関連付けられているかを理解するのに役立ちます。

theory math logic system

Illustration ideals - Image 1

noun

イデアル

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. "

数学者は数のイデアルを研究し、要素の追加や削減が常に同じコレクションの別のメンバーになるコレクションに焦点を当てました。

Illustration ideals - Image 1

noun

イデアル

(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. "

回転する物体の動きを記述するリー代数内では、イデアルは物体が回転できる特定の方法を表し、他の可能な物体の動きと組み合わせても、それらの特定の回転のいずれかになります。