Ideals with Coprime Radicals are Coprime

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

Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals.

Let their radicals be coprime:
 * $\operatorname{Rad} \left({\mathfrak a}\right) + \operatorname{Rad} \left({\mathfrak b}\right) = \left({1}\right)$

Then $\mathfrak a$ and $\mathfrak b$ are coprime.

Proof
We have:

By Unit Ideal iff Radical is Unit Ideal:
 * $\mathfrak a + \mathfrak b = \left({1}\right)$

That is, $\mathfrak a$ and $\mathfrak b$ are coprime.