Abbreviation: PIDom

A \emph{principal ideal domain} is an integral domains $\mathbf{R}=\langle R,+,-,0,\cdot,1\rangle$ in which

every ideal is principal: $\forall I \in Idl(R)\ \exists a \in R\ (I=aR)$

Ideals are defined for commutative rings

Example 1: ${a+b\theta | a,b\in Z, \theta=\langle 1+ \langle-19\rangle^{1/2}\rangle/2}$ is a Principal Ideal Domain that is not an Euclidean domains

See Oscar Campoli's “A Principal Ideal Domain That Is Not a Euclidean Domain” in <i>The American Mathematical Monthly</i> 95 (1988): 868-871

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &0 \end{array}$

Euclidean domains

Unique factorization domains