Euclidean domain

Euclidean ring is a technical term from mathematics and refers to a ring in which a ( generalized) Division is present with rest, as we know them from the integers. It is " radical" is defined by a suitable scoring function.

Definitions

There are in the literature and in academic and scientific practice, a number of different, but similar definitions of a Euclidean ring. Often it already contain specific properties, which can bring relief in the formulation of the plane defined hereinafter theory, for example. However, all of these definitions variants have in common that in a Euclidean ring a division with remainder and hence a Euclidean algorithm for finding the greatest common divisor ( gcd ) of two ring elements is possible. From this property, the name is derived.

Variant 1

An integral domain ( also referred to as integral domain, ie a commutative, zero-divisor -free ring with 1 ) is called a Euclidean ring, if an evaluation function with the following properties exist:

  • For all elements with existing ( Division of residue), wherein either or and
  • For always applies.

The evaluation function then also means Euclidean norm function ( Euclidean amount) of the ring.

Variant 2

The above definition is almost equivalent to the following, also frequently used, but in which an additional assessment for the zero is specified.

Definition: An integral domain is called Euclidean ring if an evaluation function exists with the following properties:

  • For all with exist elements ( division with remainder ), where is, and
  • For always applies.

Variant 3

Another variant provides the following

Definition: An integral domain (in this case only: a commutative, zero-divisor -free ring with at least one nonzero element ) is called a Euclidean ring if a degree function exists with the following properties:

  • For all with exist elements ( division with remainder ), where either or.

Option 3 is only supposedly weaker. In fact, the following applies: If there is integrity on a ring (with 1 ) a review of the three functions mentioned above, there are also evaluation functions that correspond to the other two definitions. It follows that the three definitions of Euclidean ring are equivalent, even though the definition of evaluation function differ.

A further substantially more general, but less widely used variant, in which the evaluation function is real-valued, but is not necessarily equivalent to the above definitions:

Variant 4

Definition: An integral domain is called Euclidean ring if a value function (or evaluation function ) exists with the following properties:

  • For all elements with existing ( Division of residue), wherein either or and
  • To a given there are at most finitely many real numbers from the range of values ​​of which are smaller than. Formal ::.

Properties

  • For evaluation functions of variants 1 and 2, where: Associated elements will be identical, in particular the units (except for the zero element ) minimum valued elements of the ring.
  • It can be shown that every Euclidean ring has a minimum evaluation function; this is from the above variant 2 There is even an algorithm to its iterative determination. However, finding a closed form for this minimal evaluation function is very expensive in general.
  • Each ring being a principal ideal Euclidean area, because if a is a minimum valued element of an ideal I, I = ( a), that is a principal ideal domain. In particular, every Euclidean ring is factorial.

Examples of Euclidean and non-Euclidean rings

  • The ring of integers is a Euclidean ring. The most natural choice for a Euclidean amount The minimum Euclidean sum of an integer is given by the length of the binary representation of its absolute value.
  • Every body K is a Euclidean ring with valuation function and for
  • The polynomial ring over a field K in a variable X is a Euclidean ring, where the Euclidean norm is given by the degree of a polynomial; this is already the minimal Euclidean norm.
  • In contrast, for example, the polynomial is not a Euclidean ring, because the ideal is not a principal ideal.
  • The ring of Gaussian numbers with the quadratic norm ( absolute value ) is a Euclidean ring.
  • The ring is not Euclidean, and since 4 does not have gcd (two " maximal common divisors " and 2, which are relatively prime however ).
  • The whole ring of the quadratic field with square -free if and only Euclidean with the quadratic norm when d is one of the following 21 numbers: -11, -7, -3, -2, -1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 d = -1 corresponds to the Gaussian numbers, d = -3 Eisenstein numbers and d = 5 the ring However, there are others, such as d = 69, for which the ring with another norm is Euclidean.

Generalization to rings with zero divisors

The definitions can be transferred to rings which are not zero divisors. Stay The above statements about the different types of definitions exist, where necessary, the inequality is to ask for. Such rings have the property that every ideal is a principal ideal, as in the zero-divisor free fall. So you are a principal ideal in a broader sense ( "principal ideal ring" or " PIR " ), but it is not a principal ideal range ( "principal ideal domain" or "PID ").

Generalization to non-commutative rings

The definitions can even be generalized to non-commutative rings, this is called the left - or rechtseuklidisch. The Hurwitzquaternionen are an example of a non-commutative ring of both the left and is also rechtseuklidisch with its standard as Euclidean norm.

Unit (ring theory) Unique factorization domain Variable (mathematics) Square-free integer Eisenstein integer Wolfram Research
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