# Principal Ideals in Integral Domain

Jump to navigation
Jump to search

## Theorem

Let $\struct {D, +, \circ}$ be an integral domain.

Let $U_D$ be the group of units of $D$.

Let $\ideal x$ be the principal ideal of $D$ generated by $x$.

Let $x, y \in \struct {D, +, \circ}$.

Then:

### Element in Integral Domain is Divisor iff Principal Ideal is Superset

- $x \divides y \iff \ideal y \subseteq \ideal x$

### Element in Integral Domain is Unit iff Principal Ideal is Whole Domain

- $x \in U_D \iff \ideal x = D$

### Elements in Integral Domain are Associates iff Principal Ideals are Equal

We are to show that:

- $x \divides y \text{ and } y \divides x \iff \ideal x = \ideal y$

Thus:

\(\ds \) | \(\) | \(\ds x \divides y \text{ and } y \divides x\) | Definition 1 of Associate in Integral Domain | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \ideal y \subseteq \ideal x \text{ and } \ideal x \subseteq \ideal y\) | Element in Integral Domain is Divisor iff Principal Ideal is Superset | |||||||||||

\(\ds \) | \(\leadstoandfrom\) | \(\ds \ideal x = \ideal y\) | Definition 2 of Set Equality |

$\blacksquare$