Euclidean domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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VIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

Definition

Symbol-free definition

An integral domain is said to be Euclidean if it admits a Euclidean norm.

Definition with symbols

An integral domain $R$ is termed a Euclidean domain if there exists a function $N$ from the set of nonzero elements of $R$ to the set of nonnegative integers satisfying the following properties:

• $N(x)=0$ if and only if $x$ is a unit
• Given nonzero $a$ and $b$ in $R$, there exist $q$ and $r$ such that $a=qb+r$ and either $r=0$ or $N(r)<N(b)$.

We call $a$ the dividend, $b$ the divisor, $q$ the quotient and $r$ the remainder.

The definition of Euclidean domain does not require that $q$ and $r$ be uniquely determined from $a$ and $b$. If $q$ and $r$ are uniquely determined from $a$ and $b$, the integral domain is termed a uniquely Euclidean domain.

Relation with other properties

Stronger properties

• Uniquely Euclidean domain
• Polynomial ring over a field

Weaker properties

• Multi-stage Euclidean domain
• Principal ideal domain
• Bezout domain
• Unique factorization domain

Metaproperties

Polynomial-closedness

This property of commutative unital rings is not closed under passing to the polynomial ring

The polynomial ring over a Euclidean domain need not be a Euclidean domain. One example is the polynomial ring with integer coefficients, which is not a Euclidean domain; another example is the polynomial ring in two variables over a field (which can be viewed as the polynomial ring in one variable, over the polynomial ring over a field).