Principal Ideals in Integral Domain

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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

$\blacksquare$