If $I,J,K$ are ideals of a ring $R$ and $I+J=J+K=K+I=R$ then $IJ+JK+KI=R$

Question

I have made some progress using the First Isomorphism theorem but would like your opinion on how this should be done

Answer 1

If $I + J = R$, then $ K \subset I + J$. Hence, $I + J + K = I + J = R$.

Answer 2

This seems pretty trivial. Of course $I+J+K\subset R$. But $R= I+J\subset I+J+K$.