$I\times J\cong R$ implies $I+J=R, I\cap J=0$?

Question

Let $R$ be a commutative ring with unity, $I,J\subset R$ ideals. Then $I\times J\cong R$ implies $I+J=R, I\cap J=0$?

The statement is clear if the isomorphism is defined by $(i,j)\mapsto i+j$, but I was wondering if it is still true with any isomorphism.