Noun

(en noun)

The act of illustrating; the act of making clear and distinct; education; also, the state of being illustrated, or of being made clear and distinct.

That which illustrates; a comparison or example intended to make clear or apprehensible, or to remove obscurity.

* {{quote-news

, year=2012 , date=June 19 , author=Phil McNulty , title=England 1-0 Ukraine , work=BBC Sport citation , page= , passage=England will regard it as a measure of justice for Frank Lampard's disallowed goal against Germany in Bloemfontein at the 2010 World Cup - but it was also an illustration of how they rode their luck for long periods in front of a predictably partisan home crowd.}}

A picture designed to decorate a volume or elucidate a literary work.

A calculated prevision of insurance premiums and returns (life insurance)"an illustration [...] is a computer projection of future premiums, cash values and death benefits based on the current dividend scale (whole life) or current interest rates and current costs of insurance (universal life)." taken from http://www.evaluatelifeinsurance.org

Noun

(en 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 -

(mathematics, order theory) A non-empty]] lower set (of a partially ordered set) which is [[closure, closed under binary suprema (a.k.a. joins).^{[http://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem#Prime_ideal_theorems]}

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 .

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

Let $\backslash mathbb\{Z\}$ be the ring of integers and let $2\backslash mathbb\{Z\}$ be its ideal of even integers. Then the quotient ring $\backslash mathbb\{Z\}\; /\; 2\backslash mathbb\{Z\}$ is a Boolean ring.

The product of two ideals $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is an ideal $\backslash mathfrak\{a\; b\}$ which is a subset of the intersection of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$. This should help to understand why maximal ideals' are prime ' ideals . Likewise, the union of $\backslash mathfrak\{a\}$ and $\backslash mathfrak\{b\}$ is a subset of $\backslash mathfrak\{a\; +\; b\}$.