2.4 PID and Euclidean domain

Some facts about PID:

• PID is a Noetherian ring.
• Bezout identity in PID: Consider $a,b\in A$ with $A$ PID. Then $(a,b)$ is also a principal ideal and so $(a,b)=(c)$ for some $c\in A$, which is unique up to a unit. We call $c$ the greatest common divisor (gcd) of $a,b$, and $c=ax+by$ for some $x,y\in A$.

Definition

An integral domain $R$ is called Euclidean if it can be equipped with a norm function $N$, where

$$N:R\setminus \{0\}\to \mathbb{N}$$

such that for any $a,b\in R$, there exists $q,r\in R$ such that $a=qb+r$ with $r=0$ or $N(r)<N(b)$. In addition, $q$ is called quotient and $r$ is called remainder.

Remark. $N$ is not necessarily multiplicative, i.e. $N(xy)$ and $N(x)N(y)$ are not always equal.

Examples.

• $R=\mathbb{Z}$ is Euclidean, where $N:a\mapsto |a|$.
• $R=\mathbb{Z}[i]=\{a+bi:a,b\in \mathbb{Z}\}$ is Euclidean, where $N:a+bi\mapsto a^2+b^2$. For any $x,y\in R$ with $x=a+bi$ and $y=c+di$. Let $x/y=z+iw$ with $z,w\in \mathbb{Q}$. Choose $\tilde{z},\tilde{w}\in \mathbb{Z}$ such that $|z-\tilde{z}|⩽1/2$ and $|w-\tilde{w}|⩽1/2$, then $q:=\tilde{z}+i\tilde{w}$ is the quotient.
• Let $k$ be a field. Then $k[x]$ and $k[[x]]$ are Euclidean, where $k[[x]]=\{{\sum }_{i=0}^{\infty }{a}_i{x}^i:a_i\in k\}$ and $N({\sum }_{i=0}^{\infty }{a}_i{x}^i)=\{i:\text{minimal }i\text{ such that }{a}_i\ne 0\}$.

Proposition

An Euclidean domain is a PID.

\begin{proof} By Euclidean algorithm. \end{proof}