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, Ockham's contribution seems to be to restrict the operation of this principle in matters pertaining to miracles and God's power: so, in the Eucharist, a plurality of miracles is possible, simply because it pleases God.

Indeed, some would criticize Melanchthon's conduct at the Diet as unbecoming of the principle he promoted, implying that faith in the truth of his cause would logically have inspired Melanchthon to a more firm and dignified posture.

Indeed, neorealists often argue that the ordering principle of the international system has not fundamentally changed from the time of Thucydides to the advent of nuclear warfare.

Indeed, he vigorously attacked rationalism, as distinguished from the rational principle, charging it with being unscientific inasmuch as it ignored the historical significance of Christianity, shut its eyes to individuality and failed to give religious feeling its due.

Indeed it was Einstein who first coined the phrase Mach's principle.

Indeed, the 1947 Constitution was authorized by the Emperor ( as was declared in the letter of promulgation ), which is in apparent conflict of the 1947 Constitution, according to which that constitution was made and authorized by the nation (" the principle of popular sovereignty ").

Indeed, sectarianism is wholly rejected: the basic principle is that all who are baptized in the name of the Trinity are Christian and form part of one church.

It is in the treatise of The Art that the most theoretical statement of this principle is to be found: ' Indeed, under a close examination spontaneity disappears ; for everything that occurs will be found to do so through something ti.

Indeed, the sides met again for ten consecutive days ( July 4 – 13 ), and a deal was reached " in principle " ( meaning the sides have agreed, but nothing is signed ) on July 13.

Indeed, there is no general agreement as to how the principle is to be interpreted, although there have been several attempts to provide formal definitions of it.

Indeed, this principle was employed the first time in the history of Structural Engineering by the Italian engineer Massimo Majowiecki, the designer of the roof of the Olympic Stadium, Rome ( built in 1990, three years before the Gottlieb-Daimler-Stadion ).

Indeed, jus cogens could be thought of as a special principle of custom with a superadded opinio juris.

Indeed, it seems that only in today's Southern fiction does Tobacco Road, with all the traditional trimmings of sowbelly and cornbread and mint juleps, continue to live -- but only as a weary, overexploited phantom.

Indeed, much apocalyptic fiction does not deal with the " end of time " but rather with the end of a certain period of time, the end of life as it is now, and the beginning of a new period of time.

Indeed, the modern Assyrian Church of the East, which reveres Nestorius, does not fully subscribe to Nestorian doctrine, though it does reject the title Theotokos.

Indeed, in most western nations relatively few physicians smoke, and their professional knowledge does appear to have a beneficial effect on their health and lifestyle.

Indeed, because it does not exist, there is no way it can be repealed, undone, overthrown, or denounced.

Indeed I wonder whether the closed and stifling atmosphere of L ' Année does not result from those contradictions.

: Indeed, Allah does not forgive association with Him, but He forgives what is less than that for whom He wills.

Indeed, in the course of the screenplay's unfolding of its events, Hud does begin to open up to his nephew.

Indeed, Ulixes does not stop his malicious gossiping until he causes Sinon's ruin with the help of the seer Calchas.

Indeed, one can choose Q to be false to express that C does not terminate.

Indeed Livia does appear somewhere ( she could hardly be excluded ), but by 13 BC Julia had politically eclipsed Livia, as has been understood and explained by many scholars.

Indeed, if a precise prescriptive definition of " heap " is available then the group consensus will always be unanimous and the paradox does not arise.

Indeed, Digital or Web reputation does not concern the virtual on-line reputation only, but the whole real reputation of a person or a company as it is affected by the Web.

Indeed, an important mantra that social scientists cite, but often forget, is that " correlation does not imply causation.

) Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.

Indeed, just because dynamical variables are " real " and not " unreal " in the metaphysical sense does not mean that they must correspond to real numbers in the mathematical sense.

Indeed, there is scarcely a document that he does not twist.

Indeed, administration of a PKMζ inhibitor into the hippocampi of rats results in retrograde amnesia with intact short-term memory ; PKMζ does not play a role in the establishment of short-term memory.

: Indeed, proof that a given condition always precedes or accompanies a phenomenon does not warrant concluding with certainty that a given condition is the immediate cause of that phenomenon.

Indeed, does not divide, does not divide, and divides for.

Indeed, Fraser cabled a senior minister, Walter Nash, after the decision was taken to accept India that " while the Declaration is not as I would have wished, it is on the whole acceptable and maximum possible, and does not at any rate leave our position unimpaired ".

' Indeed, the narrator does apologise for this explicit description, addressing the pilgrims saying: ' Ladyes, I prey yow that ye be nat wrooth ;

" Indeed for many years it was illegal in Britain to hold public office whilst believing in transubstantiation, as under the Test Act of 1673.

Indeed a small number of practising solicitors hold no degree whatsoever.

Indeed, cannabis necessarily implies to hold, and thus to buy or to produce, leaving free the judge to penalize the user on the basis of the Code of Public Health ( use ) or the Penal Code ( possession / trafficking / production ).

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold.

Indeed, as that solution preserves the well-known Heteroptera at the taxonomic rank they traditionally hold while making them a good monophyletic group, it seems preferable to the paraphyletic " Heteroptera " used in older works.

Indeed, in some common law jurisdictions, a promise by the shopkeeper to hold a specific radio would create a binding contract, even if B had to go for the money.

Indeed, since the usual practice in dealing with anticipated shot displacement cause by the wind is to " hold off " ( i. e., use a different aiming point ), some feel that the use of a return-to-battery rifle can be a disadvantage.

Indeed many other GAA grounds around the country have started to erect floodlights as the organisation starts to hold games in the evenings, whereas traditionally major matches were played almost exclusively on Sunday afternoons.

Indeed despite holding Chinese at bay throughout the morning and afternoon, the increasing difficulty of resupply and casualty evacuation made it clear that the Australians would be unable to hold Hill 504 for another night in its exposed and isolated positions.

Indeed, Democrats continued to hold most of the area's seats in the state legislature well into the 1980s.

Indeed, the PVV was floated several times as a potential coalition member by several informateurs throughout the government-formation process, and the final minority coalition under Mark Rutte between Rutte's People's Party for Freedom and Democracy and the Christian Democratic Appeal was officially " condoned " by the PVV ( although it did not hold seats in the cabinet, the PVV agreed not to bring down the government, but did so anyway in 2012 ).

Indeed, the objective was to protest against the FIFA decision to discontinue the use of La Paz as a location to hold international football matches, because of its very high elevation.

Indeed, Garvin would hold the title for 2 months before losing it back to Flair at Starrcade.

Indeed, few other Teamster big wigs even pretend that the chunky, amiable Hoffa right bower has the capacity to hold the union together for long.