Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

The main concept, however, common to all idealist epistemologies is the centrality of Reason: ( i. e.: ' Reason ' with a capital ' R '): a priori Reason: Knowledge can only be, ultimately, a product of the mind and is therefore, by definition, ' ideal '.

However, if I is infinite and the rings R < sub > i </ sub > are non-zero, then the converse is false ; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the R < sub > i </ sub >.

** Dedekind domain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals

Boyle's law shows that, at constant temperature, the product of an ideal gas's pressure and volume is always constant.

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

For this reason sweet potato vine is ideal for use in home aquariums, trailing out of the water with it's roots submerged, as it's rapid growth is fueled by toxic ammonia and nitrates, a waste product of aquatic life, which it removes from the water.

This is considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome wagered on.

where P is the pressure of the gas, V is the volume of the gas, n is the amount of substance of gas ( also known as number of moles ), T is the temperature of the gas and R is the ideal, or universal, gas constant, equal to the product of Boltzmann's constant and Avogadro's constant.

The article includes comments from Don Chadwick, who says he wasn't hired to design the ideal product for an eight-hour workday ; he was hired to update Herman Miller's previous best-seller.

While the process resulted in an extremely economical, quick and inexpensive product ( thus making it ideal for television ), it had a fatal flaw that prevented it from being taken seriously: the process involved inserting the moving lips of the voice actor over a still frame of a character's mouth.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.

The set Frac ( R ) of all fractional ideals endowed with the above product is a commutative semigroup and in fact a monoid: the identity element is the fractional ideal R.

It was shown that while rings of algebraic integers do not always have unique factorization into primes ( because they need not be principal ideal domains ), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals ( that is, every ring of algebraic integers is a Dedekind domain ).

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

While there may be theoretically ' ideal ' market segments, in reality every organization engaged in a market will develop different ways of imagining market segments, and create product differentiation strategies to exploit these segments.

Temperature consistency across a griddle surface can be tightly controlled using this type of thermostat, making it ideal for high production environments where product uniformity and consistency are critical, as much for ease of operation as food quality.

Since the product of an element in I with an element in R is in I ( as I is an ideal ), this product expression will be in I, and then ( a + b )< sup > n + m − 1 </ sup > is in I, therefore a + b is in the radical of I.

** for his fundamental and broad contributions to discrete optimization including his deep research on balanced and ideal matrices, perfect graphs and cutting planes for mixed-integer optimization.

The earliest was the bow ard, which consists of a draft-pole ( or beam ) pierced by a thinner vertical pointed stick called the head ( or body ), with one end being the stilt ( handle ) and the other a share ( cutting blade ) that was dragged through the topsoil to cut a shallow furrow ideal for most cereal crops.

Instead of stepping down in fine-grained increments to approximate a surface, the workpiece is rotated to make the cutting surfaces of the tool tangent to the ideal part features.

For many metal cutting applications the ideal has long been high-pressure, high-volume pumping to force a stream of liquid ( usually an oil-water emulsion ) directly into the tool-chip interface, with walls around the machine to contain the splatter and a sump to catch, filter, and recirculate the fluid.

The delivery of the aerosol is directly through the flutes of the tool ( it arrives directly through or around the insert itself — an ideal type of cutting fluid delivery that traditionally has been unavailable outside of a few contexts such as gun drilling or expensive, state-of-the-art liquid delivery in production milling ).

It is an " ideal " low-pass filter in the frequency sense, perfectly passing low frequencies, perfectly cutting high frequencies ; and thus may be considered to be a brick-wall filter.

While the realities of fluid dynamics dictate that a teardrop shape is the ideal aerodynamic form, Kamm found that by cutting off / flattening the streamlined end of the tear at an intermediate point, and bringing that edge down towards the ground, he could gain most of the benefit of the teardrop shape without incurring such a large material, structural, and size problem.

Informal demonstrations of these ideal characteristics amongst ranch cowboys and vaqueros evolved into the sport of reining, as well as the related events of cutting and working cow horse as well as several other horse show classes.

Walker in the mid-1960s expanded on traditional coal waterjet cutting to determine ideal nozzle shape for high-pressure waterjet cutting of stone, and Norman Franz in the late 1960s focused on waterjet cutting of soft materials by dissolving long chain polymers in the water to improve the cohesiveness of the jet stream.

For Plato, `` imitation '' is twice removed from reality, being a poor copy of physical appearance, which in itself is a poor copy of ideal essence.

In short, the fictional private eye is a specialized version of Adam Smith's ideal entrepreneur, the man whose private ambitions must always and everywhere promote the public welfare.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Again, Henley's attitude of defiance which colors his ideal of self-mastery is far from characteristic of a Stoic thinker like Marcus Aurelius, whose gentle acquiescence is almost Christian, comparable to the patience expressed in Milton's sonnet on his own blindness.

But the most fundamental objection he has to poets appears in the Tenth Book, and it is derived from his doctrine of ideal forms.

In Plato's mind there is an irresolvable conflict between the poet and the philosopher, because the poet imitates only particular objects and is incapable of rising to the first level of abstraction, much less the highest level of ideal forms.

True reality, of course, is the ideal, and the poet knows nothing of this ; ;

Even in such technical curricula as engineering, the senior is much more likely than the freshman to choose, as an ideal, liberal education over specific vocational preparation.

The old ideal of the independent entrepreneur is extant -- but so is the recognition that the main chance may be in a corporate bureaucracy.

On the other hand, the bright vision of the future has been directly stated in science fiction concerned with projecting ideal societies -- science fiction, of course, is related, if sometimes distantly, to that utopian literature optimistic about science, literature whose period of greatest vigor in the late nineteenth and early twentieth centuries produced Edward Bellamy's Looking Backward and H. G. Wells's A Modern Utopia.

But the only ideal he can think of is `` Sales ''!!

Many of these aspects will be seen as comparable to those of the ideal detective, but where the detective is active and militant, the jazz musician is passive, almost a victim of society.

Old Chris is my ideal.

The ideal girl -- possessed of talent, poise, intelligence, personality and beauty of face and figure -- is chosen each year to represent Rhode Island.

Minnesota, fabled land of waters, is in itself, ideal vacationland, having within its borders 10,000 lakes!!

The great absorbency of this tissue and the fact that it is easier to control than a sponge makes it an ideal tool for the watercolorist.

It is appropriate to call attention to certain thermodynamic properties of an ideal gas that are analogous to rubber-like deformation.

The internal energy of an ideal gas depends on temperature only and is independent of pressure or volume.

In other words, if an ideal gas is compressed and kept at constant temperature, the work done in compressing it is completely converted into heat and transferred to the surrounding heat sink.