## IDEAL

### Ideal

In ring theory, a branch of abstract algebra, an**ideal**is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal.

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### IDEAL

**iDEAL**is an e-commerce payment system used in the Netherlands, based on online banking. Introduced in 2005, this payment method allows customers to buy on the Internet using direct online transfers from their bank account.

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### ideal

#### Noun

- 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*-

- A non-empty lower set (of a partially ordered set) which is closed under binary suprema (a.k.a. joins).
^{1}- If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an
**ideal**.

- If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an
- A subring closed under multiplication by its containing ring.
- Let <math>\mathbb{Z}</math> be the ring of integers and let <math>2\mathbb{Z}</math> be its
**ideal**of even integers. Then the quotient ring <math>\mathbb{Z} / 2\mathbb{Z}</math> is a Boolean ring. - The product of two ideals <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is an ideal <math>\mathfrak{a b}</math> which is a subset of the intersection of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math>. This should help to understand why maximal
**ideals**are prime**ideals**. Likewise, the union of <math>\mathfrak{a}</math> and <math>\mathfrak{b}</math> is a subset of <math>\mathfrak{a + b}</math>.

- Let <math>\mathbb{Z}</math> be the ring of integers and let <math>2\mathbb{Z}</math> be its

#### Adjective

- Optimal; being the best possibility.
- Perfect, flawless, having no defects.
- Pertaining to ideas, or to a given idea.
- Existing only in the mind; conceptual, imaginary.
- Teaching or relating to the doctrine of idealism.
*the***ideal**theory or philosophy

- Not actually present, but considered as present when limits at infinity are included.
*ideal**point**An***ideal**triangle in the hyperbolic disk is one bounded by three geodesics that meet precisely on the circle.

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