English Dictionary

ideal

名詞

A perfect standard of beauty, intellect etc., or a standard of excellence to aim at. / (algebra, ring theory) A subring closed under multiplication by its containing ring.

日本語の意味

理想、目標とすべき完璧な美しさや知性等の標準。 / 環論において、含む環による乗法で閉じた部分環。

For many artists, the classical statue represents the ideal they strive to emulate.

多くの芸術家にとって、その古典的な彫像は彼らが目指して模倣しようとする理想像を表している。

意味（1）

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

意味（2）

(algebra, ring theory) A subring closed under multiplication by its containing ring.

Let ℤ be the ring of integers and let 2ℤ be its ideal of even integers. Then the quotient ring ℤ/2ℤ is a Boolean ring.

意味（3）

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

An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.

意味（4）

(set theory) 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 of a given set X is a nonempty subset of the powerset 𝒫(X) such that: (1)∅∈I, (2)A∈I and B⊆A⟹B∈I and (3)A,B∈I⟹A∪B∈I.

意味（5）

(algebra, 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.

If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.

意味（6）

(algebra) 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.

ideals （ plural ）