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 $\backslash mathfrak\{a\; +\; b\}$.

Noun

(en noun)

Mental]] apprehension of whatever may be known, [[think, thought, or imagined; idea, concept.

* (Isaac Newton) (1642-1727)

What hath been generally agreed on, I content myself to assume under the notion of principles.

* (George Cheyne) (1671-1743)

Few agree in their notions about these words.

* (Isaac Watts) (1674-1748)

That notion of hunger, cold, sound, color, thought, wish, or fear which is in the mind, is called the "idea" of hunger, cold, etc.

* (Alexander Hamilton) (ca.1756-1804)

Notion , again, signifies either the act of apprehending, signalizing, that is, the remarking or taking note of, the various notes, marks, or characters of an object which its qualities afford, or the result of that act.

A sentiment; an opinion.

* (Joseph Addison) (1672-1719)

The extravagant notion they entertain of themselves.

* (John Henry Newman) (1801-1890)

A perverse will easily collects together a system of notions to justify itself in its obliquity.

*{{quote-book, year=1935, author= George Goodchild

, title=Death on the Centre Court, chapter=1 , passage=“Anthea hasn't a notion in her head but to vamp a lot of silly mugwumps. She's set her heart on that tennis bloke

(label) Sense; mind. Shakespeare.

(label) An invention; an ingenious device; a knickknack.

Any small article used in sewing and haberdashery, such as a button or zipper.

(label) Inclination; intention; disposition.