Principal Ideal Domains

Principal ideal domains are integral domains where every ideal is generated by a single element, providing the natural setting for unique factorization and the structure theory of finitely generated modules.

Definition and Examples

Definition5.2Principal ideal domain (PID)

An integral domain $R$ is a principal ideal domain (PID) if every ideal $I \subseteq R$ is principal: $I = (a) = Ra$ for some $a \in R$ .

ExamplePIDs and non-PIDs

$\mathbb{Z}$ is a PID (every ideal is $(n)$ for some $n$ ).
Any field $k$ is a PID (ideals are $(0)$ and $(1) = k$ ).
$k[x]$ for $k$ a field is a PID (by the division algorithm).
$\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z}\}$ (Gaussian integers) is a PID (it is a Euclidean domain with norm $N(a+bi) = a^2 + b^2$ ).
$\mathbb{Z}[x]$ is not a PID: the ideal $(2, x) = \{f \in \mathbb{Z}[x] : f(0) \text{ even}\}$ is not principal.
$k[x,y]$ is not a PID: $(x,y)$ is not principal.

Properties

Theorem5.2Properties of PIDs

In a PID $R$ :

Every nonzero prime ideal is maximal.
$R$ is a Noetherian ring (every ascending chain of ideals stabilizes).
$R$ is a unique factorization domain (UFD).
Any two elements have a greatest common divisor: $\gcd(a,b) = d$ where $(a,b) = (d)$ .

ExampleGCD in PIDs

In $\mathbb{Z}$ : $(12, 18) = (6)$ , so $\gcd(12, 18) = 6$ . The ideal equation encapsulates the Bezout identity: $6 = 12(-1) + 18(1)$ .

In $k[x]$ : $\gcd(x^3 - 1, x^2 - 1) = x - 1$ , computed by the Euclidean algorithm.

In $\mathbb{Z}[i]$ : $\gcd(3+i, 1+2i)$ can be computed using the Gaussian integer Euclidean algorithm.

The Hierarchy: ED, PID, UFD

RemarkThe ring hierarchy

The inclusions are:

$\text{Euclidean domains (ED)} \subsetneq \text{PIDs} \subsetneq \text{UFDs} \subsetneq \text{integral domains}$

Each inclusion is strict:

$\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID but not a Euclidean domain.
$\mathbb{Z}[x]$ is a UFD (by Gauss's lemma) but not a PID.
$\mathbb{Z}[\sqrt{-5}]$ is an integral domain but not a UFD ( $6 = 2 \cdot 3 = (1+\sqrt{-5})(1-\sqrt{-5})$ ).