nLab Euclidean ring

Idea

In the same vein that commutative rings are to integral domains and GCD rings are to GCD domains, Euclidean rings are to Euclidean domains.

Definition

A Euclidean ring is a commutative ring $R$ for which there exists a function $d \colon \mathrm{Reg}(R) \to \mathbb{N}$ from the multiplicative submonoid of regular elements in $R$ to the natural numbers, often called a degree function, such that for all $f \in R$ and $g \in \mathrm{Reg}(R)$, there exist $q, r \in R$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$. There is no uniqueness requirement for $q, r$.

One could posit, instead of the property that for all $f \in R$ and $g \in \mathrm{Reg}(R)$, there exist $q, r \in R$ such that $f = q \cdot g + r$ and either $r = 0$ or $d(r) \lt d(g)$; that the commutative ring has the structure of a function $(-)\div(-):R \times \mathrm{Reg}(R) \to R$ called the division function, and a function $(-)\ \%\ (-):R \times \mathrm{Reg}(R) \to R$ called the remainder function, such that for all $f \in R$ and $g \in \mathrm{Reg}(R)$, $f = (f \div g) \cdot g + (f\ \%\ g)$ and either $f\ \%\ g = 0$ or $d(f\ \%\ g) \lt d(g)$. There might be multiple such division and remainder functions for the commutative ring $R$.

One could also add the requirement that $d(a) \leq d(a b)$ for all nonzero $a, b$. There is no loss of generality in assuming it; every Euclidean ring admits such a degree function $d'$, defining $d'(a) = \min \{d(a b): b \in \mathrm{Reg}(R)\}$.

In constructive mathematics

The definition of the Euclidean ring further bifurcates in constructive mathematics due to the disjunctive condition above. These statements, which are equivalent in the presence of excluded middle, include:

• $(r = 0) \vee (d(r) \lt d(g))$
• $\not((r \neq 0) \wedge (d(r) \geq d(g)))$
• $(d(r) \geq d(g)) \to (r = 0)$

or equivalently,

• $(f\ \%\ g = 0) \vee (d(f\ \%\ g) \lt d(g))$
• $\not((f\ \%\ g \neq 0) \wedge (d(f\ \%\ g) \geq d(g))$
• $(d(f\ \%\ g) \geq d(g)) \to (f\ \%\ g = 0)$

Properties

Every Euclidean ring is a Bézout ring, and every prefield is an Euclidean ring.

Examples

• The integers $\mathbb{Z}$ are a Euclidean ring.

• A classical Euclidean domain $A$ is a Euclidean ring where $\mathrm{Reg}(A)$ is the multiplicative monoid of elements not equal to zero (where inequality is denial inequality)

$$\mathrm{Reg}(A) \cong \{x \in A \vert x \neq 0\}$$

• A Heyting Euclidean domain $A$ is a Euclidean ring where $\mathrm{Reg}(A)$ is the multiplicative monoid of elements apart from zero

$$\mathrm{Reg}(A) \cong \{x \in A \vert x \# 0\}$$

• An ordered Euclidean domain $A$ is a Euclidean ring where $\mathrm{Reg}(A)$ is the multiplicative monoid of elements with a positive absolute value

$$\mathrm{Reg}(A) \cong \{x \in A \vert 0 \lt \vert x \vert\}$$

• The trivial ring $0$ is the unique Euclidean ring up to unique isomorphism such that $0 \in \mathrm{Reg}(0)$. The trivial ring is also the terminal Euclidean ring.

See also

• commutative ring

• GCD ring

• Bézout ring

• prefield ring

• Euclidean domain