In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ...

More precisely, a Euclidean domain is an integral domain D on which one can define a function v mapping nonzero elements of D to non-negative integers that satisfies the following division-with-remainder property: In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... Partial plot of a function f. ... The integers are commonly denoted by the above symbol. ...

• If a and b are in D and b is nonzero, then there are q and r in D such that a = bq + r and either r = 0 or v(r) < v(b).

The function v is called a valuation or norm or gauge and the key point here is that the remainder r has v-size smaller than the v-size of the divisor b. The operation mapping (a, b) to (q, r) is called the Euclidean division, whereas q is called the Euclidean quotient. In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ...

Nearly all algebra textbooks which discuss Euclidean domains include the following extra property in the definition: for all nonzero a and b in D, v(ab) ≥ v(a). This property does not have to be assumed since it is not needed to prove the most basic facts about Euclidean domains (see below). However, this inequality can always be arranged to occur by changing the choice of v, as follows: if (D,v) is a Euclidean domain as given above then the function w defined on nonzero elements of D by w(a) = least value of v(ax) as x runs over nonzero elements of D also makes D a Euclidean domain according to the above definition and it satisfies w(ab) ≥ w(a) for all nonzero a and b in D.

Examples of Euclidean domains include:

• Z, the ring of integers. Define v(n) = |n|, the absolute value of n.
• Z[i], the ring of Gaussian integers. Define v(a+bi) = a²+b², the norm of the Gaussian integer a+bi.
• K[X], the ring of polynomials over a field K. For each nonzero polynomial f, define v(f) to be the degree of f.
• K[[X]], the ring of formal power series over the field K. For each nonzero power series f, define v(f) as the degree of the smallest power of X occurring in f.
• Any discrete valuation ring. Define v(x) to be the highest power of the maximal ideal M containing x (equivalently, to the power of the generator of the maximal ideal that x is associated to). The case K[[X]] is a special case of the above.
• Any field. Define v(x) = 1 for all nonzero x.

The examples of polynomial and power series rings in one variable are the reason that the function v in the definition of a Euclidean domain is not assumed to be defined at 0. The integers are commonly denoted by the above symbol. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, a discrete valuation ring (DVR) is a particular kind of commutative ring that is a local ring, which satisfies conditions that in algebraic geometry come from non-singularity of a point on an algebraic curve. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

Every Euclidean domain is a principal ideal domain. In fact, if I is a nonzero ideal of a Euclidean domain D and a nonzero a in I is chosen to minimize v(a) over all elements of I, then I = aD. The proof of this does not use the inequality v(ab) ≥ v(a). In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

The name Euclidean domain comes from the fact that the extended Euclidean algorithm can be carried out in any Euclidean domain. The proof that this algorithm terminates does not use the inequality v(ab) ≥ v(a). The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ...

In order to prove every nonzero nonunit in a Euclidean domain is a product of irreducibles, the inequality v(ab) ≥ v(a) is useful for an inductive argument. Or one could instead appeal to the proof of this same result for any principal ideal domain (or Noetherian domain) and once again avoid having to use the inequality v(ab) ≥ v(a). In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...