Ideals with Coprime Radicals are Coprime
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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.
$\blacksquare$