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:

\(\ds \map \Rad {\mathfrak a + \mathfrak b}\)

\(=\)

\(\ds \map \Rad {\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} }\)

Radical of Sum of Ideals

\(\ds \)

\(=\)

\(\ds \map \Rad {\ideal 1}\)

$\map \Rad {\mathfrak a}$ and $\map \Rad {\mathfrak b}$ are coprime

\(\ds \)

\(=\)

\(\ds \ideal 1\)

Radical of Unit Ideal

By Unit Ideal iff Radical is Unit Ideal:

$\mathfrak a + \mathfrak b = \ideal 1$

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

$\blacksquare$