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:
 * $\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} = \ideal 1$

Then $\mathfrak a$ and $\mathfrak b$ are coprime:
 * $\mathfrak a + \mathfrak b = \ideal 1$

Proof
We have:

By Unit Ideal iff Radical is Unit Ideal:
 * $\mathfrak a + \mathfrak b = \ideal 1$

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