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

(order of senses) (en-noun)

The state or quality of being true to someone or something.

(label) Faithfulness, fidelity.

* (Samuel Taylor Coleridge) (1772-1834)

Alas! they had been friends in youth, / But whispering tongues can poison truth .

(label) A pledge of loyalty or faith.

True facts, genuine depiction or statements of reality.

* (Samuel Taylor Coleridge) (1772-1834)

The truth depends on, or is only arrived at by, a legitimate deduction from all the facts which are truly material.

*{{quote-magazine, date=2014-06-21, volume=411, issue=8892, magazine=(The Economist)

, title= Magician’s brain , passage=The truth is that [Isaac] Newton was very much a product of his time. The colossus of science was not the first king of reason, Keynes wrote after reading Newton’s unpublished manuscripts. Instead “he was the last of the magicians”.}}

Conformity to fact or reality; correctness, accuracy.

* {{quote-magazine, date=2012-01, author=Robert M. Pringle, volume=100, issue=1, page=31, magazine=(American Scientist), title= How to Be Manipulative

, passage=As in much of biology, the most satisfying truths in ecology derive from manipulative experimentation. Tinker with nature and quantify how it responds.}}

Conformity to rule; exactness; close correspondence with an example, mood, model, etc.

* John Mortimer (1656?-1736)

Ploughs, to go true, depend much on the truth of the ironwork.

That which is real, in a deeper sense; spiritual or ‘genuine’ reality.

* 1820 , (John Keats), (Ode on a Grecian Urn)

Beauty is truth', ' truth beauty, - that is all / Ye know on earth, and all ye need to know.

(label) Something acknowledged to be true; a true statement or axiom.

* 1813 , (Jane Austen), (Pride and Prejudice)

It is a truth universally acknowledged, that a single man in possession of a good fortune, must be in want of a wife.

Topness. (See also truth quark.)