(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If A and B are ideals of a commutative ring R with unity and A + B = R, show that [itex]A \cap B[/itex] = AB.

3. The attempt at a solution

Showing [itex]AB \subseteq A \cap B[/itex] is easy. I'm having trouble with containment in the other direction:

Let [itex]x \in A \cap B[/itex]. Then x is in A and x is in B. To show that x belongs to AB, it suffices to show that 1 belongs to either A or B and so 1x or x1 belongs to AB. It seems to me that 1 isn't necessarily in A or B so this approach is unfruitful. Is there another decomposition of x into ab where a belongs to A and B belongs to B?

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# Homework Help: A Rings and Ideals Problem

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