PID Implies UFD

Statement

A principal ideal domain (PID) is an integral domain in which every ideal is principal (generated by a single element). A unique factorisation domain (UFD) is an integral domain in which every non-zero non-unit factors into irreducibles uniquely up to order and associates.

Theorem. Every PID is a UFD.

Examples: $\mathbb{Z}$, $k[x]$ for a field $k$, and the Gaussian integers $\mathbb{Z}[i]$ are PIDs (hence UFDs).

Visualization

The Gaussian integers $\mathbb{Z}[i]$ form a PID. Factor $2 \in \mathbb{Z}[i]$:

2 = (1+i)(1-i)
  = -i · (1+i)^2          [since 1-i = -i(1+i)]

Norm check: $N(1+i) = 1^2 + 1^2 = 2$, so $N(2) = 4 = N(1+i)^2$. The element $1+i$ is irreducible in $\mathbb{Z}[i]$ (its norm $2$ is prime in $\mathbb{Z}$).

Factorisation	Notes
$2 = (1+i)(1-i)$	both irreducible, norms $= 2$
$1-i = -i(1+i)$	$-i$ is a unit (norm $1$)
so $2 = -i(1+i)^2$	unique up to units

Any other factorisation of $2$ in $\mathbb{Z}[i]$ is obtained by multiplying factors by units $\{1, -1, i, -i\}$.

Proof Sketch

1. Every irreducible is prime. In a PID, if $p \mid ab$ and $p \nmid a$, then $(p, a) = (d)$ for some $d$. Since $p$ is irreducible, $d$ is a unit, so $(p, a) = R$. Write $1 = rp + sa$, then $b = rpb + sab$; since $p \mid ab$, we get $p \mid b$. So $p$ is prime.

2. Existence of factorisations. A PID is Noetherian (all ideals are finitely generated — trivially, by one element). In a Noetherian domain, every non-zero non-unit has an irreducible factorisation: if $a$ is not irreducible, write $a = bc$ and continue; this cannot go on indefinitely by the ascending chain condition on ideals $(a) \subset (b) \subset \cdots$.

3. Uniqueness. Since every irreducible is prime (step 1), uniqueness of factorisation follows by the standard argument: if $p_1 \cdots p_r = q_1 \cdots q_s$, then $p_1$ divides some $q_j$; since $q_j$ is irreducible, $p_1$ and $q_j$ are associates. Cancel and induct.

Connections

The chain Euclidean domain $\to$ PID $\to$ UFD is the backbone of commutative algebra; see Euclidean DomainEuclidean Domain$$\forall a, b \ne 0,\; \exists q, r:\; a = qb + r,\; N(r) < N(b)$$A domain with a division algorithm: any a, b give a = qb + r with r smaller than b under a norm function.Read more → for the first implication. UFD structure is what makes Gauss's Lemma (Polynomial)Gauss's Lemma (Polynomial)$$f,g \text{ primitive} \implies fg \text{ primitive}$$The product of primitive polynomials is primitive; irreducibility over a UFD transfers to irreducibility over its fraction field.Read more → work: the content map is multiplicative precisely when $R$ is a UFD.

-- Mathlib: IsPrincipalIdealRing.to_uniqueFactorizationMonoid
-- in Mathlib.RingTheory.PrincipalIdealDomain (line 344)
/-- Every PID is a UFD.
    Mathlib provides this as an automatic instance. -/
example (R : Type*) [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] :
    UniqueFactorizationMonoid R :=
  inferInstance