Radical of Unit Ideal

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

Let $\ideal 1$ be its unit ideal.

Then its radical equals $\ideal 1$:
 * $\map \Rad {\ideal 1} = \ideal 1$.

Proof
By definition of ideal:
 * $\map \Rad {\ideal 1} \subseteq A$

By Ideal of Ring is Contained in Radical:
 * $\ideal 1 = A \subseteq \Rad {\ideal 1}$.

By definition of set equality:
 * $\map \Rad {\ideal 1} = \ideal 1$