I have some problem to understanding the proof of this problem. This theorem is on page $414$ introduction to homological algebra Rotman. The theorem says:
If $R$ is a domain, then $\operatorname{Tor}_n^R(A,B)$ is a torsion module for all $A$, $B$ and $\forall n\ge 1$.
for proof we use the $$0\to {tB}\to B \to B/tB\to0$$
gives exactness of $$ Tor_1^R(A,tB)\to Tor_1^R(A,B)\to Tor_1^R(A,B/tB).$$the flanking terms are torsion, thus $Tor_1^R(A.B)$ is torsion(?!) and says the proof by dimension shifting(?)
I have thought about this, but I don't know how to use dimension shifting!
Can you help please?
thank you