Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?

Question

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$.

This is very easy to prove if the ring is unital as you may write $1=i+j$, and then $m\otimes n=m1\otimes n=mi\otimes jn=0$. But what about the case when $R$ is not unital, is it still true then?