In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination
of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
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.
Euclidean domains appear in the following chain of class inclusions:
specifically in ring theory, a Euclideandomain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows...
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...
two numbers Euclideandomain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and...
Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclideandomains and all...
denote a specific function from the domain to the natural numbers called a "Euclidean function". Although "Euclidean division" is named after Euclid, it...
Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely...
Eisenstein integers of norm 1. The ring of Eisenstein integers forms a Euclideandomain whose norm N is given by the square modulus, as above: N ( a + b ω...
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...
many properties with integers: they form a Euclideandomain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization...
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest...
elements Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal Euclideandomain, an integral domain which allows...
is a Euclideandomain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. The...
and more generally this is true in GCD domains. If R is a Euclideandomain in which euclidean division is given algorithmically (as is the case for instance...
polynomial ring R[x] is a principal ideal domain and, more importantly to our discussion here, a Euclideandomain. It can be shown that the degree of a polynomial...
{\displaystyle \mathbb {Z} } is a Euclideandomain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer can...
real quadratic integers that is a principal ideal domain is also a Euclideandomain for some Euclidean function, which can indeed differ from the usual...
group over a field or a Euclideandomain is generated by transvections, and the stable special linear group over a Dedekind domain is generated by transvections...
Euclidean division similar to that of integers. Every Euclideandomain is a principal ideal domain, and thus a UFD. In a Euclideandomain, Euclidean division...
rings for which such a theorem exists are called Euclideandomains. Like for the integers, the Euclidean division of the polynomials may be computed by...
Similarly, the domain of a left Euclidean relation is a subset of its range, and the restriction of a left Euclidean relation to its domain is an equivalence...
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclideandomain. Analogously,...
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