Radical of Unit Ideal

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

Let $(1)$ be its unit ideal.

Then its radical equals $(1)$:
 * $\sqrt{(1)} = (1)$.

Proof
By definition of ideal, $\sqrt{(1)} \subseteq A$.

By Ideal of Ring is Contained in Radical, $(1) = A \subseteq \sqrt{(1)}$.

By definition of set equality, $\sqrt{(1)} = (1)$.