Direct sum of non-zero ideals over an integral domain

Question

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$

I regarded the short exact sequence $0\to I\cap J\to I\oplus J\to I+J\to0$ with the maps $(\cdot,\;-\,\cdot)$ and $\mathrm{pr}_1+\mathrm{pr}_2$. It splits if $I+J$ is a principal ideal and $I\oplus J\cong (I+J)\oplus(I\cap J)\cong R\oplus(I\cap J)$. In general, however, this is not the case. I cannot find any counter-example, either.

Answer 1

In this answer you can find two principal ideals $I,J$ such that $I\cap J$ is not finitely generated. Then you can't have $R\oplus (I\cap J)\simeq I\oplus J$ since then $R\oplus (I\cap J)\simeq R\oplus R$, and thus $I\cap J$ is isomorphic to a quotient module of $R\oplus R$ and therefore it's finitely generated.

Answer 2

If $R$ is a commutative ring, and $L\subset R$ is an ideal such that $R\oplus L\simeq R^2$ then $L\simeq R$.

We have $\bigwedge^2(R\oplus L)\simeq\bigwedge^2R^2$, so $L\simeq R$. (Here I've used that $L$ is a projective rank one module, and this gives us $\bigwedge^2L=0$.)

This shows that the example suggested by Crostul works: let $R=\mathbb Z[\sqrt{-5}]$, $I=(2)$, and $J=(1+\sqrt{-5})$. If $R\oplus(I\cap J)\simeq I\oplus J$ then $R\oplus(I\cap J)\simeq R\oplus R$. We have to prove that $I\cap J$ is not principal, and this is left to the reader.