## Investigate the kernel of a module homomorphism to a given tensor product

Let $R$ be an integral domain with field of fractions $Q$ and let $N$ be a left unital $R$-module. Prove that the kernel of the $R$-module homomorphism $\iota : N \rightarrow Q \otimes_R N$ given by $n \mapsto 1 \otimes n$ is $\mathsf{Tor}(N)$.

If $n \in \mathsf{ker}\ \iota$, then $1 \otimes n = 0$. By a previous exercise, there exists a nonzero $r \in R$ such that $r \cdot n = 0$. Thus $n \in \mathsf{Tor}_R(N)$.

Conversely, suppose $n \in \mathsf{Tor}_R(N)$ with $r \in R$ nonzero such that $r \cdot n = 0$. Then $\iota(n) = 1 \otimes n = 0$, again using the previous exercise.