Euclidean domains, PIDs, and UFDs form a hierarchy of rings where divisibility behaves increasingly like divisibility in \(\mathbb Z\). The key theme is when ideals, greatest common divisors, and factorizations are controlled by single elements.
Divisibility, units, and associates
Definition. In an integral domain \(R\), one writes \(a\mid b\) if \(b=ac\) for some \(c\in R\). A unit is an invertible element. Elements \(a,b\) are associates if \(a=ub\) for a unit \(u\).
Definition. A nonzero nonunit \(p\) is irreducible if \(p=ab\) forces \(a\) or \(b\) to be a unit. It is prime if \(p\mid ab\) implies \(p\mid a\) or \(p\mid b\).
Proposition. Prime elements are irreducible in any integral domain.
Euclidean domains
Definition. A Euclidean domain is an integral domain \(R\) equipped with a function \(\delta:R\setminus\{0\}\to\mathbb N\) such that for \(a,b\in R\), \(b\ne0\), there exist \(q,r\in R\) with \[ a=qb+r, \qquad r=0\text{ or }\delta(r)<\delta(b). \]
Example. \(\mathbb Z\) is Euclidean by absolute value, and \(F[x]\) is Euclidean by degree. The Euclidean algorithm computes greatest common divisors in both rings.
Principal ideal domains
Theorem. Every Euclidean domain is a PID: every ideal is generated by one element.
Proof. Let \(I\ne0\). Choose \(b\in I\) with \(\delta(b)\) minimal. For any \(a\in I\), divide \(a=qb+r\). Since \(r=a-qb\in I\), minimality forces \(r=0\), hence \(a\in(b)\). Thus \(I=(b)\).
Unique factorization domains
Definition. A UFD is an integral domain in which every nonzero nonunit factors into irreducibles, and the factorization is unique up to order and associates.
Theorem. Every PID is a UFD.
Proof. Existence follows by ruling out an infinite strictly increasing chain of principal ideals \((a_1)\subsetneq(a_2)\subsetneq\cdots\). For uniqueness, one proves irreducible elements are prime in a PID using Bezout identities, then cancels prime factors inductively.
Strictness of the hierarchy
Theorem. There is a chain of implications \[ \text{Euclidean domain}\Rightarrow\text{PID}\Rightarrow\text{UFD}, \] and neither converse holds in general.
Example. \(F[x,y]\) is a UFD but not a PID, since the ideal \((x,y)\) is not principal. Some rings of algebraic integers are PIDs but not Euclidean, showing the first converse can fail as well.
A nonunique factorization warning
Counterexample. In \(\mathbb Z[\sqrt{-5}]\), \[ 6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5}) \] gives two essentially different factorizations. This ring is not a UFD.