Radical of Unit Ideal

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

Let $\left({1}\right)$ be its unit ideal.

Then its radical equals $(1)$:
 * $\operatorname{Rad} \left({\left({1}\right)}\right) = \left({1}\right)$.

Proof
By definition of ideal:
 * $\operatorname{Rad} \left({\left({1}\right)}\right) \subseteq A$

By Ideal of Ring is Contained in Radical:
 * $\left({1}\right) = A \subseteq \operatorname{Rad} \left({\left({1}\right)}\right)$.

By definition of set equality:
 * $\operatorname{Rad} \left({\left({1}\right)}\right) = \left({1}\right)$