ideals background

ideals

/aɪˈdɪəlz/ /aɪˈdi.əlz/

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noun

理想，抱负

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. "

即使很困难，玛丽亚仍然努力在日常行为中践行她诚实和善良的理想。

value philosophy moral quality attitude being character

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noun

理想环，理想子环

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. "

在抽象代数中，“理想环”（或称“理想子环”）是环内一种特殊的子集，它具有“吸收”环中其他元素乘法的性质。

Illustration ideals - Image 1

noun

理想，下集，并闭集

(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. "

在格理论中研究理想（一种特殊的集合，满足下集和并闭集的性质），有助于数学家理解元素的集合是如何根据特定的包含和组合规则组织和关联的。

theory math logic system

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noun

理想集合, 理想集

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. "

数学家研究了数的理想集，专注于那些加减元素后，结果仍然属于同一集合的集合群。

Illustration ideals - Image 1

noun

李代数理想，理想子代数

(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. "

在描述旋转物体运动的李代数中，理想子代数代表物体旋转的特定方式，这些旋转方式与物体的任何其他可能运动相结合后，结果仍然是这些特定旋转方式之一。