# Definition of Euclidean Domain and Principal Ideal Domain (PID)

A Euclidean domain is an integral domain $R$ with a function $d:R\setminus \{0\}\to \mathbb{N}$ satisfying the following:

(1) $d(a)\leq d(ab)$ for all nonzero $a,b$ in $R$.

(2) for all $a,b \in R$, $b\neq 0$, $\exists q, r, \in R$ such that $a=bq+r$, with either $r=0$ or $d(r).

(d is known as the Euclidean function)

On the other hand, a Principal ideal domain (PID) is an integral domain in which every ideal is principal (can be generated by a single element).

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