Wish I had known this in 2012

Let A be an integral domain, I,J\subseteq A ideals such that I+J=A. If I\cap J is principal, then I and J are projective.

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Wish I had known this in 2012

2 thoughts on “Wish I had known this in 2012

  1. The sequence 0\rightarrow I\cap J \rightarrow I\oplus J\rightarrow A\rightarrow 0, where the last arrow is (i,j)\mapsto i-j, is exact by hypothesis. Now I\cap J is principal and therefore isomorphic to A, since A is a domain, and the sequence splits, since the rightmost term is projective. Then I\bigoplus J\simeq A\bigoplus A and so both ideals are projective.

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