Bézout's lemma is true in any principal ideal domain, but there are integral domains in which it is not true.

An integral domain in which Bézout's identity holds is called a Bézout domain.

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

The same construction can be generalized to the field of fractions of any integral domain.

The integers are an example of an integral domain which does not allow all divisions as, again, whole numbers are needed.

The entire functions on the complex plane form an integral domain ( in fact a Prüfer domain ).

So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID.

Let R be an integral domain.

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function.

As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following ( not exhaustive ) chain of class inclusions:

One way to find that analytic continuation is to use Euler's integral for positive arguments and extend the domain to negative numbers by repeated application of the recurrence formula,

If R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

The following is an example of an integral domain with two elements that do not have a gcd:

In abstract algebra, an integral domain is a commutative ring that has no zero divisors, and which is not the trivial ring

It is another integral operator ; it is useful mainly because it converts a function on one ( temporal ) domain to a function on another ( frequency ) domain, in a way effectively invertible.

** A variable for a 2-dimensional region in calculus, usually corresponding to the domain of a double integral.

If R is an integral domain and f and g are polynomials in R, it is said that f divides g or f is a divisor of g if there exists a polynomial q in R such that f q = g. One can show that every zero gives rise to a linear divisor, or more formally, if f is a polynomial in R and r is an element of R such that f ( r ) = 0, then the polynomial ( X − r ) divides f. The converse is also true.

* Principal ideal domain, in abstract algebra, an integral domain in which every ideal is principal

* An ideal I in the ring R is prime if and only if the factor ring R / I is an integral domain.

In particular, a commutative ring is an integral domain if and only if

Next I refer to our program in space exploration, which is often mistakenly supposed to be an integral part of defense research and development.

the Athletic program at Carleton is considered an integral part of the activities of the College and operates under the same budgetary procedure and controls as the academic work.

Indeed, a study of the individual child is an integral part of the work of the elementary-school teacher, rather than merely an additional chore.

What Parker and his contemporaries -- Gillespie, Davis, Monk, Roach ( Tristano is an anomaly ), etc. -- did was to absorb the musical ornamentation of the older jazz into the basic structure, of which it then became an integral part, and with which it then developed.

Within only a few years, foamed plastics materials have managed to grow into an integral, and important, phase of the plastics industry -- and the end is still not yet in sight.

As in the theory of perception, established in psycho-physiology, the eye is recognized as an integral part of the brain.

As retinal images are conceded to be an integral function of the brain it seems logical to suppose that the nerves, between the inner brain and the eyes, carry the direct drive for cooperation from the various brain centers -- rather than to theorize on the transmission of an image which is already in required location.

The truth, however, is that the ecumenical church is just the local church in its own true character as an integral unit of the whole People of God throughout the world.

In Continental philosophy ( particularly phenomenology and existentialism ), there is much greater tolerance of ambiguity, as it is generally seen as an integral part of the human condition.

He is the so-called " Mad Arab " credited with authoring the imaginary book Kitab al-Azif ( the Necronomicon ), and as such is an integral part of Cthulhu Mythos lore.

With over 120 million visitors a year tourism is integral to the Alpine economy with much it coming from winter sports although summer visitors are an important component of the tourism industry.

* The Aluk religion in the Toraja society and the people of Tana Toraja, embrace religious rituals such as the funeral ceremony where a sacred cockfight, known as bulangan londong or saung, is an integral part of the ceremony and considered sacred because of the spilling of blood on the earth in spiritual appeasement.

Of particular interest to 20th-century music theorists is the attention he paid to silence as an integral part of music.

is the phase integral ( integration of reflected light ; a number in the 0 to 1 range ).

As this tension is an integral part of AA, Rudy and Greil argue that AA is best described as a quasi-religious organization.

Lobster is an integral ingredient to the cuisine, indigenous to the coastal waters of the region.

A metal case holds an integral primer to initiate the propellant and provides the gas seal to prevent the gases leaking out of the breech, this is called obturation.

However, because each schema object is integral to the definition of Active Directory objects, deactivating or changing these objects can fundamentally change or disrupt a deployment.

Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.

Because of this, each of the infinitely many antiderivatives of a given function f is sometimes called the " general integral " or " indefinite integral " of f and is written using the integral symbol with no bounds:

In the latter example the ring can be made into an UFD by taking its integral closure in ( the ring of Dirichlet integers ), over which becomes reducible, but in the former example R is already integrally closed.