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.
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?