Category:Euclidean Domains

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This category contains results about Euclidean Domains.
Definitions specific to this category can be found in Definitions/Euclidean Domains.

Let $\struct {R, +, \circ}$ be an integral domain with zero $0_R$.

Let there exist a mapping $\nu: R \setminus \set {0_R} \to \N$ with the properties:

$(1): \quad$ For all $a, b \in R, b \ne 0_R$, there exist $q, r \in R$ with $\map \nu r < \map \nu b$, or $r = 0_R$ such that:
$a = q \circ b + r$
$(2): \quad$ For all $a, b \in R, b \ne 0_R$:
$\map \nu a \le \map \nu {a \circ b}$

Then $\nu$ is called a Euclidean valuation and $R$ is called a Euclidean domain.