 Solution

Work modulo $I$. Putand let $A$ be the image of $I^{\prime }/I$ in $C$. The separating condition $I^{\prime }=I^{\prime }\cap (I+J)$ modulo $I$ says that $I^{\prime }/I\to C$ is injective, so we may regard $A$ as a submodule of $C$.

Since $r\in \sqrt{(I:I^{\prime })}$, some power $r^q$ annihilates $A$. Apply the Artin-Rees lemma to $A\subseteq C$ and the principal ideal $(r)$. There is $s$ such that for every $m\ge s$,For $m\ge s+q$, the right side is zero. Pulling the equality $A\cap r^{m}C=0$ back to $R$ givesas required.

Solved by gpt-5.6-sol high.