Noun

(en noun)

The quality (positive or negative) that renders something desirable or valuable.

* {{quote-news, year=2012, date=May 13, author=Alistair Magowan, work=BBC Sport

, title= Sunderland 0-1 Man Utd , passage=United were value for their win and Rooney could have had a hat-trick before half-time, with Paul Scholes also striking the post in the second half.}}

The degree of importance given to something.

* {{quote-magazine, date=2013-06-07, author=(Gary Younge)

, volume=188, issue=26, page=18, magazine=(The Guardian Weekly) , title= Hypocrisy lies at heart of Manning prosecution , passage=WikiLeaks did not cause these uprisings but it certainly informed them. The dispatches revealed details of corruption and kleptocracy that many Tunisians suspected, […]. They also exposed the blatant discrepancy between the west's professed values and actual foreign policies.}}

The amount (of money or goods or services) that is considered to be a fair equivalent for something else.

* M'Culloch

An article may be possessed of the highest degree of utility, or power to minister to our wants and enjoyments, and may be universally made use of, without possessing exchangeable value .

* Dryden

His design was not to pay him the value of his pictures, because they were above any price.

* {{quote-magazine, date=2013-08-03, volume=408, issue=8847, magazine=(The Economist)

, title= Boundary problems , passage=Economics is a messy discipline: too fluid to be a science, too rigorous to be an art. Perhaps it is fitting that economists’ most-used metric, gross domestic product (GDP), is a tangle too. GDP measures the total value of output in an economic territory. Its apparent simplicity explains why it is scrutinised down to tenths of a percentage point every month.}}

(music) The relative duration of a musical note.

(arts) The relative darkness or lightness of a color in (a specific area of) a painting etc.

* Joe Hing Lowe

I establish the colors and principal values by organizing the painting into three values--dark, mediumand light.

Numerical quantity measured or assigned or computed.

Precise meaning; import.

the value''' of a word; the '''value of a legal instrument

(Mitford)

(obsolete) Esteem; regard.

(Dryden)

* Bishop Burnet

My relation to the person was so near, and my value for him so great.

(obsolete) valour; also spelled valew

(Spenser)

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