Unit Ideal iff Radical is Unit Ideal

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

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

Then:
 * $ \mathfrak a = A \iff \map \Rad {\mathfrak a} = A$

where:
 * $A$ is called the unit ideal of $A$
 * $\map \Rad {\mathfrak a}$ denotes the radical of $\mathfrak a$