A binary relation is the special case of an n-ary relation R ⊆ A < sub > 1 </ sub > × … × A < sub > n </ sub >, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A < sub > j </ sub > of the relation.

This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval

An important result on the topology of R < sup > n </ sup >, that is far from superficial, is Brouwer's invariance of 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.

However, if there is no " obvious " Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

Let R be an integral domain.

However, one can show that ( EF2 ) is superfluous in the following sense: any domain R which

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

Let R be a domain and f a Euclidean function on R. Then:

* R is a principal ideal domain.

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.

However if R is a unique factorization domain, then any two elements have a gcd, and more generally this is true in gcd domains.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If R is commutative, then one can associate to every polynomial P in R, a polynomial function f with domain and range equal to R ( more generally one can take domain and range to be the same unital associative algebra over R ).

His sailing vessel is guided by fate to the shores of his own country at a time when Sibylla's domain is overrun by the armies of one of her rejected suitors.

In this domain the simple fact of coexistence in the same local, national, and world community is enough to guarantee that we cannot refrain from having some effect, large or small, upon Gentile-Jewish relations.

The difference is important, for although the older law of nations did cover relationships among sovereigns, this was by no means its exclusive domain.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

This exon encodes a portion of the mature TNF domain, as well as the leader sequence which is a highly conserved region necessary for proper intracellular processing.

The protein is organized with 6 transmembrane segments, then the C1 cytoplasmic domain, then another 6 membrane segments, and then a second cytoplasmic domain called C2.

In Mycobacterium tuberculosis, the AC-III polypeptide is only half as long, comprising one 6-transmembrane domain followed by a cytoplasmic domain, but two of these form a functional homodimer that resembles the mammalian architecture.

Abbadie gave his domain the name Abbadia, which is the name still used in Basque.

The castle was classified as a protected historical monument by France in 1984 and most of the domain now belongs to the Coastal Protection Agency and is managed by the city of Hendaye.

It is often used in signal processing for analyzing functions or series of values, such as time domain signals.

Active Directory ( AD ) is a directory service created by Microsoft for Windows domain networks.

For example, when a user logs into a computer that is part of a Windows domain, Active Directory checks the submitted password and determines whether the user is a system administrator or normal user.

A tree is a collection of one or more domains and domain trees in a contiguous namespace, linked in a transitive trust hierarchy.

If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.

The sets X and Y are called the domain ( or the set of departure ) and codomain ( or the set of destination ), respectively, of the relation, and G is called its graph.

As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

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.

In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i. e., can be generated by a single element.

A field norm is a Dedekind-Hasse norm ; thus, ( 5 ) shows that a Euclidean domain is a PID.

A Bézout domain is thus a GCD domain, and ( 4 ) gives yet another proof that a PID is a UFD.

An immediate consequence of the definition is that every principal ideal domain ( PID ) is a Dedekind domain.

In fact a Dedekind domain is a unique factorization domain ( UFD ) if and only if it is a PID.

By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust.

But a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring ( DVR ), so the same local characterization cannot hold for PIDs: rather, one may say that the concept of a Dedekind ring is the globalization of that of a DVR.

Zariski and Samuel were sufficiently taken by this construction to pose as a question whether every Dedekind domain arises in such a fashion, i. e., by starting with a PID and taking the integral closure in a finite degree field extension.

Thus these are precisely the class of domains for which Frac ( R )/ Prin ( R ) forms a group, the ideal class group Cl ( R ) of R. This group is trivial if and only if R is a PID, so can be viewed as quantifying the obstruction to a general Dedekind domain being a PID.

In view of the well known and exceedingly useful structure theorem for finitely generated modules over a principal ideal domain ( PID ), it is natural to ask for a corresponding theory for finitely generated modules over a Dedekind domain.

In abstract algebra, a discrete valuation ring ( DVR ) is a principal ideal domain ( PID ) with exactly one non-zero maximal ideal.

In mathematics, the Smith normal form is a normal form that can be defined for any matrix ( not necessarily square ) with entries in a principal ideal domain ( PID ).