Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$.

Question

Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$.

I attempted proving this fact using the definition of what it means to be a torsion module and the definition of $Tor_n(X,Y)$ using a projective resolution of $X$, but it doesn't come easily due to the difference in dimensions. How should this fact be shown?

Answer 1

Take a projective resolution of $Y$. Then, tensoring with $X$ yields only torsion modules and thus the subquotients (in particular, the homology groups of the complex, aka the various $\mathrm{Tor}$ modules) are still torsion.