Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power

Theorem
Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$.

Let $a \in D$ be a proper element of $D$.

Then:
 * $\forall n \in \Z_{\ge 0}: \ideal {a^{n + 1} } \subsetneq \ideal {a_n}$

where $\ideal x$ denotes the principal ideal of $D$ generated by $x$.

Proof
We have:

It remains to be shown that $\ideal {a^{n + 1} } \ne \ideal {a^n}$.

$\ideal {a^{n + 1} } = \ideal {a^n}$.

Then:

That is, $a$ is a unit of $D$.

This contradicts the assertion that $a$ is a proper element of $D$.

The result follows by Proof by Contradiction.