$I+J=R$ and $IJ$ is principal. prove that $I$ is a projective module.

Question

let $I$ and $J$ be ideals of a commutative ring with identity $R$.
assume that $I+J=R$ and that $IJ$ is principal.
prove that $I$ is a projective module.

I am familiar tiwh the theorem that states that if $M,N$ are modules, $N$ is free and $\alpha:M\rightarrow N$ is a surjective homomorphism then $N\oplus $ker$(\alpha)$ is isomorphic to $M$

I tried defining $\beta : I\oplus J\rightarrow R$ with $(a,b)\mapsto a+b$ and conclude that $R\oplus$ker$(\beta)$ is isomorphic to $I\oplus J$, yet I haven't manged to prove that ker$(\beta)$ is free.