Table of Contents
Principal Ideal Domain
Abbreviation: PIDom
Definition
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
Morphisms
Examples
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
Basic results
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &1
f(5)= &1
f(6)= &0
\end{array}$
