Suppose $R$ is a Ring and $B$ is a $R-$ module. Define $_{r}R=\left\{s\in S | rs=0 \right\}$. If $_{r}R$ is not zero, we have an exact sequence: $$ 0\rightarrow _{r}R \rightarrow R \xrightarrow {\text{times r}} R \rightarrow R/rR \rightarrow 0$$. Prove that we have an exact sequence:$$0\rightarrow Tor_{2}^{R}(R/rR,B)\rightarrow _{r}R\otimes B\rightarrow _{r}B \rightarrow Tor_{1}^{R}(R/rR,B)\rightarrow 0$$ and $Tor_{n}^{R}(R/rR,B)\cong Tor_{n-2}^{R}(_{r}R,B)$ for $n>3$.
Things are quite simple when $_{r}R=0$, because the first exact sequence would become projective resolution of $R/rR$. But now it's not projective resolution. From the outlook of the result we have to prove, it seems to me that we have to find new projective resolutions for both $R/rR$ and $R/rR$ which are somehow related to the given exact sequence. I had a hard time finding projective resolutions for these two objects.