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

Given: Integers $a,b\in\mathbb Z$ with $b\ne0$.

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 $1\ne0$ (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 $1\ne0$ and no zero divisors is an integral domain (Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors).

For $b\ne0$, there are $q,r\in\mathbb Z$ with $a=qb+r$ and $0\le r<|b|$ (Division with remainder for any nonzero divisor: for $a \in \mathbb{Z}$ and $b \ne 0$ there are unique $q, r \in \mathbb{Z}$ with $a = qb + r$ and $0 \le r < |b|$).

Integer absolute values are nonnegative, vanish exactly at zero, and satisfy $|r|=r$ for $r\ge0$ (Absolute value in $\mathbb{Z}$: $|a| \ge 0$; $|a| = 0$ exactly when $a = 0$; $|-a| = |a|$; $|ab| = |a|\,|b|$; $-|a| \le a \le |a|$; and $|a| \le c$ exactly when $-c \le a \le c$, The absolute value $|a|$ 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).