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Euclidean domain (ED)

Definition

A Euclidean domain R is an integral domain where there is a function $N:R\setminus\{0\} \to \mathbb{N}$ such that given any $a, b \in R$ where $b \ne$ there exist $q,r \in R$ such that $a = b.q + r$ and either $r=0$ or $N(r)<N(b)$.

Properties

A matrix over a euclidean domain can be written in Smith normal form.

Euclidean domains are PIDs and therefore also Integral Domains

Examples

The integers $\mathbb{Z}$ and Gaussian integers $\mathbb{Z}[i] are Euclidean domains.$

The ring of polynomials $\mathbb{F}[t]$ over a field $\mathbb{F}$ is a Euclidean domain.

all these pages adapted with probably insufficient credit from my university's lecture notes