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 |
$\blacksquare$