### 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}$