In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

Although a combination treatment program may be ideal, any of the treatments can be done individually.

These are the results that the Library of Congress expected of an ideal mass deacidification treatment in 1994.

Numerous research studies since the 1950s have focused on the ideal treatment for diabetic ketoacidosis.

Preventive dental treatment is also necessary ( and often overlooked by the patient ), as the lack of saliva associated with xerostomia creates an ideal environment for the proliferation of bacteria that cause dental caries ( cavities ).

" In his campaign for humane treatment of the insane he approached Couthon the leader of the Paris Commune who agreed to visit Biutre because he believed strongly in the ideal of equality of human beings.

Because many business and government policies were designed to accommodate the " ideal worker " ( that is, the traditional male worker who had no such responsibilities ) rather than caregiver-workers, inefficient and inequitable treatment has resulted.

Completely stopping the use of alcohol is the ideal goal of treatment.

The other state, Luzania, constitutes the most significant treatment of the topic of an " ideal state " in Lem ( and many would argue, in all science fiction ).

It is widely accepted that corticosteroid treatment after a Kasai operation, with or without choleretics and antibiotics, has a beneficial effect on the postoperative bile flow and can clear the jaundice ; but the dosing and duration of the ideal steroid protocol have been controversial (" blast dose " vs. " high dose " vs. " low dose ").

In ideal circumstances, patients with known varices should receive treatment to reduce their risk of bleeding.

The ideal treatment for IgAN would remove IgA from the glomerulus and prevent further IgA deposition.

The diagnosis of hypopituitarism is made by blood tests, but often specific scans and other investigations are needed to find the underlying cause, such as tumors of the pituitary, and the ideal treatment.

The growing amount of recidivism in the latter half of the nineteenth century led a number of criminologists to argue that " imprisonment did not, and could not fulfill its original ideal of treatment aimed at reintegrating the offender into the community ".

Oral naltrexone remains an ideal treatment only for a small part of the opioid-dependent population, usually the ones with an unusually stable social situation and motivation ( e. g., dependent health care professionals ).

The novel is the most thorough treatment of the theme of Waugh's writing, first fully displayed in Brideshead Revisited: a celebration of the virtues of tradition, of family and feudal loyalty, of paternalist hierarchy, of the continuity of institutions and of the heroic ideal and the calamitous disappearance of these which has led to the emptiness and futility of the modern world.

The ideal NNT is 1, where everyone improves with treatment and no one improves with control.

Patients whose outward drift occurs at distance rather than at near distance are less ideal candidates for treatment.

The theoretical treatment of devices such as capacitors and resistors tends to assume they are ideal or " perfect " devices, contributing only capacitance or resistance to the circuit.

The ideal treatment for anemia of chronic disease is to treat the chronic disease successfully, but this is rarely completely possible with current medical treatment.

When the hypothalamus receives signals from one of its many inputs ( e. g., cerebral cortex, limbic system, visceral organs ) about conditions that deviate from an ideal homeostatic state ( e. g., alarming sensory stimulus, emotionally charged event, energy deficiency ), this can be interpreted as the initiation step of the stress-response cascade.

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.

From the outset, this new republic held that film would be the most ideal propaganda tool for the Soviet Union because of its widespread popularity among the established citizenry of the new land ; Vladimir Lenin, in fact, declared it the most important medium for educating the masses in the ways, means and successes of Communism, a position which was later echoed by Joseph Stalin.

Among the most important figures of the federal era include the first democratically elected president in Honduras, Dionisio de Herrera, a lawyer, whose government, begun in 1824 established the first constitution, Gen. Francisco Morazán, Federal President 1830-1834 and 1835 – 1839, whose figure embodies the ideal American Unionist, and José Cecilio del Valle, editor of the Declaration of Independence signed in Guatemala on September 15, 1821 and Foreign Minister of Mexico in 1823.

; Pharmacology: While it is well established that the skin provides an ideal site for the administration of local and systemic drugs, it presents a formidable barrier to the permeation of most substances.

Lenin and a small group of anti-war socialist leaders, including Rosa Luxemburg and Karl Liebknecht, denounced established socialist leaders of having betrayed the socialist ideal via their support of the war.

In the Republic, the ideal community is, “... a city which would be established in accordance with nature .”

Once the Metastasian ideal had been firmly established, comedy in Baroque-era opera was reserved for what came to be called opera buffa.

Modern Architecture, as established and developed by people such as Walter Gropius, Le Corbusier, and Philip Johnson, was focused on the pursuit of a perceived ideal perfection, and attempted harmony of form and function, and dismissal of " frivolous ornament.

The Whig canon and the neo-Harringtonians, John Milton, James Harrington and Sidney, Trenchard, Gordon and Bolingbroke, together with the Greek, Roman, and Renaissance masters of the tradition as far as Montesquieu, formed the authoritative literature of this culture ; and its values and concepts were those with which we have grown familiar: a civic and patriot ideal in which the personality was founded in property, perfected in citizenship but perpetually threatened by corruption ; government figuring paradoxically as the principal source of corruption and operating through such means as patronage, faction, standing armies ( opposed to the ideal of the militia ), established churches ( opposed to the Puritan and deist modes of American religion ) and the promotion of a monied interest — though the formulation of this last concept was somewhat hindered by the keen desire for readily available paper credit common in colonies of settlement.

Habermas argued that the public sphere was bourgeois, egalitarian, rational, and independent from the state, making it the ideal venue for intellectuals to critically examine contemporary politics and society, away from the interference of established authority.

True families fulfilling God's ideal will be begun, and the Kingdom of God will be established both on earth and in heaven.

During his administration ( 1870 – 1882 ), the College achieved higher academic standing and established an ideal model for intellectual honesty and sound scholarship that gained national recognition.

" It established a strategic model he believed to be ideal — the use of defensive tactics within a strategic offensive.

In ideal situations, no antimicrobial therapy should be administered until microbiological microscopy and culture studies have established the specific infection involved.

Caesar is representative of the ideal king, who brings about the Pax Romana similar to the political peace established under the Tudors.

By the 19th Century BCE, Harran was established as a merchant outpost due to its ideal location.

When a prison camp was established in Ojus in the early 1900s, rock mining in the area began and it was discovered that the rock coming out of Ojus had qualities that made it ideal for road building.

After hosting the 1939 Golden Gate International Exposition, the United States Navy deemed Treasure Island an ideal location for transporting people and machines to the Pacific theater, and on April 1, 1941, established Treasure Island Naval Station which also included a portion of Yerba Buena Island.

A Sloan Foundation grant established the MIT School of Industrial Management in 1952 with the charge of educating the " ideal manager ", and the school was renamed in Sloan's honor as the Alfred P. Sloan School of Management, one of the world's premier business schools.

At Welney, these washes provide an ideal habitat for wildfowl and other birds, and the Wildfowl and Wetlands Trust has established a nature reserve here, the WWT Welney.

Such was the ideal which Wolf had in his mind when he established the philological seminarium at Halle.

" The idea, therefore, was already established by this time that natural beauty becomes true beauty only when modified in accordance with a human ideal.

By this Polykleitos meant that a statue should be composed of clearly definable parts, all related to one another through a system of ideal mathematical proportions and balance, no doubt expressed in terms of the ratios established by Pythagoras for the perfect intervals of the musical scale: 1: 2 ( octave ), 2: 3 ( harmonic fifth ), and 3: 4 ( harmonic fourth ).

True families fulfilling God's ideal will be begun, and the Kingdom of God will be established both on earth and in heaven.