Ideal is Unit Ideal iff Includes Unity

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

Let $\mathfrak a$ be an ideal of $A$.

Then:
 * $\mathfrak a = A \iff 1 \in \mathfrak a$

where $1$ denotes the unity of $A$.

Proof
$\implies$ is trivial.

To see $\impliedby$, suppose $1 \in \mathfrak a$.

Let $a \in A$ be arbitrary.

Then: