# If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals

## Problem 360

Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have
$I_1+I_2=R.$

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.

## Proof.

Since $I_1+I_2=R$, there exists $a \in I_1$ and $b \in I_2$ such that
$a+b=1.$ Then we have
\begin{align*}

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