Unit Ideal is Principal Ideal Generated by Unity

Theorem
Let $A$ be a commutative ring with unity.

Then:
 * $A = \ideal 1$

where:
 * $A$ is called the unit ideal of $A$
 * $\ideal 1$ denotes the principal ideal generated by the unity of $A$

Proof
$\ideal 1 \subseteq A$ is clear by definition of principal ideal.

To see $A \subseteq \ideal 1$, let $a \in A$ be an arbitrary element.

Then: