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.

Indeed, quinine is the only drug recommended by the WHO as first-line treatment for uncomplicated malaria in pregnancy.

Indeed, Phocion met Nicanor, requesting a mild treatment for the Athenians.

Indeed, mental treatment to cure a workaholic can successfully reduce the hours spent on the job, while increasing the person's productivity.

Indeed, he established the form almost single-handedly and, through innovative and imaginative treatment of standard dance forms of the time, paved the way for Johann Sebastian Bach's elaborate contributions to the genre ( not to mention almost every major composer in Europe, since the vast majority composed suites and were influenced by the " French style " exemplified by Froberger ).

The Vita tells that Dagobert I sent his representative the merciless general Lollo ( Lollonius ) to reside at Bourges and bring the city more closely under the king's command ; when the people came complaining of their treatment to Sulpicius, he decreed a three-day fast for clergy and laity, but also sent one of his clergy Ebargisilus by name to Clovis II with a remonstrance that alarmed the young king to the extent that he revoked the census or tax, and returned the assessment, setting a precedent, it was claimed: " Indeed, having been relieved from affliction, those people remain in total liberty up to this present day.

" ( 92 ) Indeed, Orinase was marketed by Upjohn not as a cure-all for all diabetics, but specifically as a treatment that was " not an oral insulin " and " did not work in all diabetics.

Indeed, fosmidomycin has been tested in combination treatment with clindamycin for treatment of malaria with favorable results.

Indeed, as already noted, the Court proceeded on the assumption that the executives involved in the dealings between Du Pont and General Motors acted `` honorably and fairly '' and exercised their business judgment only to serve what they deemed the best interests of their own companies.

Indeed, if pressed, we would say what the late Robert Henri, American painter, said to a pupil, `` Anything will do for a subject: it's what you do with it that counts ''.

Indeed, the format is ideally suited to such analysis because the viewer is afforded the opportunity to see how a person reacts on the outside through the constant recording of their actions and also what they feel on the inside through the Diary Room / Confession Room.

" Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.

Indeed, even the Greenwich meridian itself is not quite what it used to be — defined by " the centre of the transit instrument at the Observatory at Greenwich ".

Indeed, the question of what is meant by a " right " is itself controversial and the subject of continued philosophical debate.

Indeed, it is the only course I could take to be faithful to what I believe.

Indeed, one-size-fits-all programming is no longer seen as tenable by some, as the diversity of musical tastes among the listening public have created a proliferation of radio formats in what some might call a form of narrowcasting.

For one example, he expressed this playfulness in what is perhaps his most famous rhyme, a twist on Joyce Kilmer's verse: " I think that I shall never see / a poem lovely as a tree ", which drops " billboard " in place of poem and adds, " Indeed, unless the billboards fall / I'll never see a tree at all.

Indeed, the position has been described as undergoing a " presidentialisation ", to the point that its incumbents publicly outshine the actual head of state ; former governor general Adrienne Clarkson alluded to what she saw as " an unspoken rivalry " that had developed between the prime minister and the Crown.

Indeed this, together with the grid pattern of the streets, is what one would expect in Prussia ’ s chief garrison city.

Indeed, such was the perceived threat of the Ottoman Empire under the reign of Suleiman that ambassador Busbecq warned of Europe's imminent conquest: " On Turks ' side are the resources of a mighty empire, strength unimpaired, habituation to victory, endurance of toil, unity, discipline, frugality and watchfulness ... Can we doubt what the result will be ?... When the Turks have settled with Persia, they will fly at our throats supported by the might of the whole East ; how unprepared we are I dare not say.

Indeed, some have described council communism as " socialism from below ," which they counterpose against what they see as the " socialism from above " that was endorsed by Stalinism.

Indeed, about ten per cent of people with dementia have what is known as " mixed dementia ", which may be a combination of Alzheimer's disease and multi-infarct dementia.

Indeed, after the Ascension of the Lord the Apostles handed on to their hearers what He had said and done.

Indeed, a growing number of language acquisition researchers argue that the very idea of a strict rule-based grammar in any language flies in the face of what is known about how languages are spoken and how languages evolve over time.

Indeed, even today, many " Baresi " are still unaware of what happened and why.

Indeed, the conservation of 4-momentum in inertial motion via curved space-time results in what we call gravitational force in general relativity theory.

Indeed our people are already migrating to escape, and are already suffering from the consequences of what world authorities on climate change have consistently been warning us.

Indeed, because his exact appearance is never directly described in Old English by the original Beowulf poet, part of the debate revolves around what is known, namely his descent from the biblical Cain ( who was the first murderer in the Bible ).

Indeed, much of what is known about the celebrations associated with Orcus come from medieval sources.

Indeed, whoever will compare what he says in of the heresy which he describes but forbears to name, with Prosper's account of the charges brought against Augustine by certain Semipelagian clergymen of Marseille, will have little doubt that Vincentius and they had the same teacher in view, and were of the same mind with regard to his teaching.

Indeed, until the ability to dial a phone number came into widespread use in the 1950s ( it had existed in limited form since the 1920s ), telephone users could not place calls without operator assistance.

Indeed, linkydinky's page on the hoax calls the report " our hoax ".

Indeed, Takato's latest scheme involves assisting a murderer who calls himself the Gankutsuō (" The Count of Monte Cristo ").

Indeed, Marx sees the essence of what he calls " the capital relation " as being summarised by the circumstance that " capital buys labour ", i. e. the power of property ownership to command human energy and labour-time, and thus of inanimate " things " to exert an autonomous power over people.

Indeed, this " musical cross-pollination ", as Vivian Host calls it, represents a local appropriation of global musical forms, such that the blending of different musics creates the music of a " new world.

This argument fails in considering the imbalance in self-reported political allegiances by journalists themselves, that distort any market analogy as regards offer: (...) Indeed, in 1982, 85 percent of Columbia Graduate School of Journalism students identified themselves as liberal, versus 11 percent conservative " ( Lichter, Rothman, and Lichter 1986: 48 ), quoted in Sutter, 2001.

Indeed, many top political leaders in Singapore come from peasant backgrounds, while modern peasants boast about their great ancestry.

Indeed the growth of a vast empire by Arab-Muslim Jihads is not devoid of economic and political issues and brought in its wake the first real growth of the slave trade.

Indeed, a person's opinions regarding the role of government often determine the rest of their political ideology.

Indeed, Taft's administration marked a change in style from the political charisma of Roosevelt to the passion of Taft for the rule of law.

Indeed these views continued throughout his life as he associated himself with other key political figures such as Eva Gore Booth.

Indeed, for Vice President Agnew it has become a code-word describing the activities of muggers, arsonists, draft evaders, campaign hecklers, campus militants, anti-war demonstrators, juvenile delinquents and political assassins.

Indeed, Escape from Freedom is viewed as one of the founding works of political psychology.

Indeed, Jiang Qing's role in the Cultural Revolution was perceived by the public to be largely negative, and the Gang of Four was a convenient scapegoat for the ten years of political and social turmoil.

Indeed, as she becomes aware of the political machinations among the older priestesses Thar and Kossil, the Tombs become a refuge to her, as she is the only one who may freely move through the labyrinth under them.

Indeed, following the 1960s, it increasingly became home to a number of squatters, artists, bohemians and also political radicals.

Indeed, he is in many circles regarded as a political opportunist ; the very model of a new breed of party members directly associated with the widespread corruption and cronyism that flourished during his tenure.

Indeed, the events of October 1970 galvanized a loss of support for the violent wing of the Quebec separatist movement that had gained support over nearly ten years, and increased support for political means of attaining independence, including support for the separatist Parti Québécois, which went on to take power at the provincial level in 1976.

Indeed, Fox's generous nature ( helped by his earlier over-indulgence by his father ) made him a rather hopeless political party-leader, often seeming quite unwilling to cajole or discipline his unruly supporters, especially in the case of the young, who were disproportionately represented among the Foxites.

Indeed, " some social theorist have argued that the late-20th century preoccupation with the self serves as a tool of social control: soothing political unrest ... one's own pursuit of self-invention.

Indeed, Kammu's avoidance of drastic reform decreased the intensity of political struggles, and he became recognized as one of Japan's most forceful emperors.

Indeed, he almost seemed to take a turn to the political Right during his final two years in the Senate, as witnessed by his opposition to President Richard Nixon's Family Assistance Plan, a form of " reverse income tax " to help the poor get off of welfare and a program similar to a plan he had proposed several years earlier — though it should be noted that many liberal senators and representatives also opposed the plan.

Indeed, much of the political tension in modern societies is the conflict between the desire to organize a nation-state using the tribal values of egalitarianism and unity and the simple fact that large societies are unavoidably impersonal and sometimes not amenable to small-society rules.

Indeed, people have been stopped at airports solely due to their political views recently ( see No-fly list ) and there appears to be little public will to stop practices of this nature.

Indeed, Strauss wrote that Heidegger's thinking must be understood and confronted before any complete formulation of modern political theory is possible.

Indeed he has Balkanised a few political parties himself "

Indeed, partly due to the political concerns in Western European and the United States about the Slavic world nurtured by the Cold War, Slavic studies flourished in the years from World War II into the 1990s and remains strong ( though university enrollments in Slavic languages have declined since the nineties ).

Indeed, no nation in the world had voter-based political parties.

Indeed, Merv was the center of Abbasid partisanship for the duration of the Abbasid revolution, and later on became a consistent source of political support for the Abbasid rulers in Baghdad, and the governorship of Khurasan at Merv was considered one of the most important political figures of the Caliphate.