How can I prove that $\text{Tor}\_1(R/I,R/J) = (I \cap J)/IJ$, where $R$ is a ring and $I, J$ ideals. 


Moreover, if we suppose $R=I+J$, how do I prove that $\text{Tor}\_1(R/I,R/J)=0$?


Ps: No, this is not a homework question. 



How to calculate Tor(R/I, R/J) ??
Hints: 1) First prove that $I\otimes(R/J)=I/IJ$ . 2) If $I+J=R$, write $1=i+j$ and use the fact that $x=1x$. 


