Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation: Examples and Counterexamples

Given: Integers $a,b\in \mathbb{Z}$ with $b\mathrm{\ne }0$.

The integers form a commutative ring (The integers form a commutative ring).

The canonical embedding $\mathbb{N}\to \mathbb{Z}$ is injective, preserves addition, multiplication and order, and has the nonnegative integers as its image; in particular it makes $1\mathrm{\ne }0$ (The naturals embed in the integers).

The product of two nonzero integers is nonzero (The integers have no zero divisors; multiplicative cancellation).

A commutative ring with $1\mathrm{\ne }0$ and no zero divisors is an integral domain (Zero divisor, and integral domain: a commutative ring with $1\mathrm{\ne }0$ and no zero divisors).

For $b\mathrm{\ne }0$, there are $q,r\in \mathbb{Z}$ with $a=qb+r$ and $0\le r<\mathrm{\mid }b\mathrm{\mid }$ (Division with remainder for any nonzero divisor: for $a\in \mathbb{Z}$ and $b\mathrm{\ne }0$ there are unique $q,r\in \mathbb{Z}$ with $a=qb+r$ and $0\le r<\mathrm{\mid }b\mathrm{\mid }$).

Integer absolute values are nonnegative, vanish exactly at zero, and satisfy $\mathrm{\mid }r\mathrm{\mid }=r$ for $r\ge 0$ (Absolute value in $\mathbb{Z}$: $\mathrm{\mid }a\mathrm{\mid }\ge 0$; $\mathrm{\mid }a\mathrm{\mid }=0$ exactly when $a=0$; $\mathrm{\mid }-a\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }$; $\mathrm{\mid }ab\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }$; $-\mathrm{\mid }a\mathrm{\mid }\le a\le \mathrm{\mid }a\mathrm{\mid }$; and $\mathrm{\mid }a\mathrm{\mid }\le c$ exactly when $-c\le a\le c$, The absolute value $\mathrm{\mid }a\mathrm{\mid }$ of an integer).

A Euclidean domain is an integral domain equipped with a natural-valued function satisfying the stated division alternative (Euclidean domain and Euclidean function).