# Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

Given an integral domain $R$, and a left torsion $R$-module $A$ (i.e. $\forall{a}\in A,\exists{r}\in R$ such that $ra=0$) how would you show that $\mathrm{Tor}_n^R(A,B)$ is also a torsion $R$-module?

#### Solutions Collecting From Web of "Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion."

Choose a projective resolution $P_\bullet \to B \to 0$. Then $\mathrm{Tor}_n(A,B) \stackrel{\mathrm{def}}{=} H_n(A \otimes P_\bullet)$. This is a quotient of a submodule of $A \otimes P_n$, so that it suffices to observe that $A \otimes P_n$ is torsion, which is obvious (if $ra=0$ then $r(a \otimes p)=0$).