How can I prove that Tor(R/I,R/J) = (I|J)/IJ, where | denotes intersection, R is a ring and I, J ideals.

Moreover, if we suppose R=I+J, how do I prove that Tor (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.