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 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. 


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 $x=1x$. 


Although already pointed out by others that this is an easy exercise, the most obvious answer to the second question has been overlooked: since $I$ and $J$ are both in the annihilator of $\text{Tor}_1(R/I,R/J)$ (by functoriality) and $I+J=R$, the Tor module must be zero. 

