Let R be a commutative ring with identity and I and J be ideals of R such that I+J=R. Let us prove that IJ=I∩J.
Since I is an ideal, IJ⊂IR⊂I. Since J is an ideal, IJ⊂RJ⊂J. Therefore, IJ⊂I∩J.
On the other hand, taking into accounr that R is a commutative ring with identity, we conclude that
I∩J⊂(I∩J)R⊂(I∩J)(I+J)=(I∩J)I+(I∩J)J⊂JI+IJ=IJ+IJ=IJ.
Consequently, IJ=I∩J.
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