# 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**.

## Subcategories

This category has only the following subcategory.

### E

## Pages in category "Euclidean Domains"

The following 11 pages are in this category, out of 11 total.

### E

- Element is Unit iff its Euclidean Valuation equals that of 1
- Elements of Euclidean Domain have Greatest Common Divisor
- Euclid's Lemma for Euclidean Domains
- Euclid's Lemma for Irreducible Elements
- Euclid's Lemma for Irreducible Elements/General Result
- Euclidean Domain is Principal Ideal Domain
- Euclidean Valuation of Non-Unit is less than that of Product