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: $\sqrt{\mathfrak a} + \sqrt{\mathfrak b} = (1)$.

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

Proof
We have:

By Unit Ideal iff Radical is Unit Ideal, $\mathfrak a + \mathfrak b = (1)$.

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