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.

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