ideal

Forms

1.Pertaining to ideas, or to a given idea.
2.Existing only in the mind; conceptual, imaginary.
- Life and death appeared to me ideal bounds, which I should first break through, and pour a torrent of light into our dark world.
- At first, he began to doubt the reality of his adventures, but the acute pain in his shoulders when he attempted to rise, assured him that the kicking of the goblins was certainly not ideal.
3.Optimal; being the best possibility.
4.Perfect, flawless, having no defects.

1.A perfect standard of beauty, intellect etc., or a standard of excellence to aim at.
- Varvara Andreevna, when I was very young, I set before myself the ideal of the woman I loved and should be happy to call my wife.
2.A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
- Definition 15.11 (Width Ideal) An ideal Q of a poset P = (X,≤) is a width ideal if maximal(Q) is a width antichain.
3.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.
4.A two-sided ideal; a subset of a ring which is closed under both left and right multiplication by elements of the ring.
- However, every R has a minimal prime ideal consisting of left zero-divisors and one of right zero-divisors.
- If an ideal I of a ring contains the multiplicative identity 1, then we have seen that I must be the entire ring.
- In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals.

Uncommon· #3139

From French idéal, from Late Latin ideālis (“existing in idea”), by surface analysis, idea + -al, from Latin idea (“idea”); see idea.